2013
2013
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Paper 1, Section I, D
2013 commentShow that for .
Let be a sequence of positive real numbers. Show that for every ,
Deduce that tends to a limit as if and only if does.
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Paper 1, Section I, F
2013 comment(a) Suppose for and . Show that converges.
(b) Does the series converge or diverge? Explain your answer.
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Paper 1, Section II, D
2013 comment(a) Determine the radius of convergence of each of the following power series:
(b) State Taylor's theorem.
Show that
for all , where
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Paper 1, Section II, E
2013 comment(a) Let . Suppose that for every sequence in with limit , the sequence converges to . Show that is continuous at .
(b) State the Intermediate Value Theorem.
Let be a function with . We say is injective if for all with , we have . We say is strictly increasing if for all with , we have .
(i) Suppose is strictly increasing. Show that it is injective, and that if then
(ii) Suppose is continuous and injective. Show that if then . Deduce that is strictly increasing.
(iii) Suppose is strictly increasing, and that for every there exists with . Show that is continuous at . Deduce that is continuous on .
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Paper 1, Section II, E
2013 comment(i) State (without proof) Rolle's Theorem.
(ii) State and prove the Mean Value Theorem.
(iii) Let be continuous, and differentiable on with for all . Show that there exists such that
Deduce that if moreover , and the limit
exists, then
(iv) Deduce that if is twice differentiable then for any
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Paper 1, Section II, F
2013 commentFix a closed interval . For a bounded function on and a dissection of , how are the lower sum and upper sum defined? Show that .
Suppose is a dissection of such that . Show that
By using the above inequalities or otherwise, show that if and are two dissections of then
For a function and dissection let
If is non-negative and Riemann integrable, show that
[You may use without proof the inequality for all .]
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Paper 2, Section I, A
2013 commentSolve the equation
subject to the conditions at .
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Paper 2, Section I, A
2013 commentUse the transformation to solve
subject to the conditions and at , where is a positive constant.
Show that when
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Paper 2, Section II, A
2013 commentThe function satisfies the equation
Give the definitions of the terms ordinary point, singular point, and regular singular point for this equation.
For the equation
classify the point according to your definitions. Find the series solution about which satisfies
For a second solution with at , consider an expansion
where and . Find and which have and . Comment on near for this second solution.
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Paper 2, Section II, A
2013 commentFind and which satisfy
subject to at .
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Paper 2, Section II, A
2013 commentMedical equipment is sterilised by placing it in a hot oven for a time and then removing it and letting it cool for the same time. The equipment at temperature warms and cools at a rate equal to the product of a constant and the difference between its temperature and its surroundings, when warming in the oven and when cooling outside. The equipment starts the sterilisation process at temperature .
Bacteria are killed by the heat treatment. Their number decreases at a rate equal to the product of the current number and a destruction factor . This destruction factor varies linearly with temperature, vanishing at and having a maximum at .
Find an implicit equation for such that the number of bacteria is reduced by a factor of by the sterilisation process.
A second hardier species of bacteria requires the oven temperature to be increased to achieve the same destruction factor . How is the sterilisation time affected?
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Paper 2, Section II,
2013 commentConsider the function
Determine the type of each of the nine critical points.
Sketch contours of constant .
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Paper 4, Section I, B
2013 commentA hot air balloon of mass is equipped with a bag of sand of mass which decreases in time as the sand is gradually released. In addition to gravity the balloon experiences a constant upwards buoyancy force and we neglect air resistance effects. Show that if is the upward speed of the balloon then
Initially at the mass of sand is and the balloon is at rest in equilibrium. Subsequently the sand is released at a constant rate and is depleted in a time . Show that the speed of the balloon at time is
[You may use without proof the indefinite integral ]
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Paper 4, Section , B
2013 commentA frame moves with constant velocity along the axis of an inertial frame of Minkowski space. A particle moves with constant velocity along the axis of . Find the velocity of in .
The rapidity of any velocity is defined by . Find a relation between the rapidities of and .
Suppose now that is initially at rest in and is subsequently given successive velocity increments of (each delivered in the instantaneous rest frame of the particle). Show that the resulting velocity of in is
where .
[You may use without proof the addition formulae and .]
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Paper 4, Section II, B
2013 comment(a) A particle of unit mass moves in a plane with polar coordinates . You may assume that the radial and angular components of the acceleration are given by , where the dot denotes . The particle experiences a central force corresponding to a potential .
(i) Prove that is constant in time and show that the time dependence of the radial coordinate is equivalent to the motion of a particle in one dimension in a potential given by
(ii) Now suppose that . Show that if then two circular orbits are possible with radii and . Determine whether each orbit is stable or unstable.
(b) Kepler's first and second laws for planetary motion are the following statements:
K1: the planet moves on an ellipse with a focus at the Sun;
K2: the line between the planet and the Sun sweeps out equal areas in equal times.
Show that K2 implies that the force acting on the planet is a central force.
Show that K2 together with implies that the force is given by the inverse square law.
[You may assume that an ellipse with a focus at the origin has polar equation with and .]
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Paper 4, Section II, B
2013 comment(a) A rigid body is made up of particles of masses at positions . Let denote the position of its centre of mass. Show that the total kinetic energy of may be decomposed into , the kinetic energy of the centre of mass, plus a term representing the kinetic energy about the centre of mass.
Suppose now that is rotating with angular velocity about its centre of mass. Define the moment of inertia of (about the axis defined by ) and derive an expression for in terms of and .
(b) Consider a uniform rod of length and mass . Two such rods and are freely hinged together at . The end is attached to a fixed point on a perfectly smooth horizontal floor and is able to rotate freely about . The rods are initially at rest, lying in a vertical plane with resting on the floor and each rod making angle with the horizontal. The rods subsequently move under gravity in their vertical plane.
Find an expression for the angular velocity of rod when it makes angle with the floor. Determine the speed at which the hinge strikes the floor.
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Paper 4 , Section II, B
2013 comment(i) An inertial frame has orthonormal coordinate basis vectors . A second frame rotates with angular velocity relative to and has coordinate basis vectors . The motion of is characterised by the equations and at the two coordinate frames coincide.
If a particle has position vector show that where and are the velocity vectors of as seen by observers fixed respectively in and .
(ii) For the remainder of this question you may assume that where and are the acceleration vectors of as seen by observers fixed respectively in and , and that is constant.
Consider again the frames and in (i). Suppose that with constant. A particle of mass moves under a force . When viewed in its position and velocity at time are and . Find the motion of the particle in the coordinates of . Show that for an observer fixed in , the particle achieves its maximum speed at time and determine that speed. [Hint: you may find it useful to consider the combination .]
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Paper 4, Section II, B
2013 comment(a) Let with coordinates and with coordinates be inertial frames in Minkowski space with two spatial dimensions. moves with velocity along the -axis of and they are related by the standard Lorentz transformation:
A photon is emitted at the spacetime origin. In it has frequency and propagates at angle to the -axis.
Write down the 4 -momentum of the photon in the frame .
Hence or otherwise find the frequency of the photon as seen in . Show that it propagates at angle to the -axis in , where
A light source in emits photons uniformly in all directions in the -plane. Show that for large , in half of the light is concentrated into a narrow cone whose semi-angle is given by .
(b) The centre-of-mass frame for a system of relativistic particles in Minkowski space is the frame in which the total relativistic 3-momentum is zero.
Two particles and of rest masses and move collinearly with uniform velocities and respectively, along the -axis of a frame . They collide, coalescing to form a single particle .
Determine the velocity of the centre-of-mass frame of the system comprising and .
Find the speed of in and show that its rest mass is given by
where
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Paper 3, Section I, D
2013 commentState Lagrange's Theorem.
Let be a finite group, and and two subgroups of such that
(i) the orders of and are coprime;
(ii) every element of may be written as a product , with and ;
(iii) both and are normal subgroups of .
Prove that is isomorphic to .
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Paper 3, Section I, D
2013 commentDefine what it means for a group to be cyclic, and for a group to be abelian. Show that every cyclic group is abelian, and give an example to show that the converse is false.
Show that a group homomorphism from the cyclic group of order to a group determines, and is determined by, an element of such that .
Hence list all group homomorphisms from to the symmetric group .
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Paper 3, Section II, D
2013 comment(a) Let be a finite group. Show that there exists an injective homomorphism to a symmetric group, for some set .
(b) Let be the full group of symmetries of the cube, and the set of edges of the cube.
Show that acts transitively on , and determine the stabiliser of an element of . Hence determine the order of .
Show that the action of on defines an injective homomorphism to the group of permutations of , and determine the number of cosets of in .
Is a normal subgroup of Prove your answer.
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Paper 3, Section II, D
2013 comment(a) Let be a prime, and let be the group of matrices of determinant 1 with entries in the field of integers .
(i) Define the action of on by Möbius transformations. [You need not show that it is a group action.]
State the orbit-stabiliser theorem.
Determine the orbit of and the stabiliser of . Hence compute the order of .
(ii) Let
Show that is conjugate to in if , but not if .
(b) Let be the set of all matrices of the form
where . Show that is a subgroup of the group of all invertible real matrices.
Let be the subset of given by matrices with . Show that is a normal subgroup, and that the quotient group is isomorphic to .
Determine the centre of , and identify the quotient group .
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Paper 3, Section II, D
2013 comment(a) Let be the dihedral group of order , the symmetry group of a regular polygon with sides.
Determine all elements of order 2 in . For each element of order 2 , determine its conjugacy class and the smallest normal subgroup containing it.
(b) Let be a finite group.
(i) Prove that if and are subgroups of , then is a subgroup if and only if or .
(ii) Let be a proper subgroup of , and write for the elements of not in . Let be the subgroup of generated by .
Show that .
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Paper 3, Section II, D
2013 commentLet be a prime number.
Prove that every group whose order is a power of has a non-trivial centre.
Show that every group of order is abelian, and that there are precisely two of them, up to isomorphism.
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Paper 4, Section I, E
2013 commentLet and be positive integers. State what is meant by the greatest common divisor of and , and show that there exist integers and such that . Deduce that an integer divides both and only if divides .
Prove (without using the Fundamental Theorem of Arithmetic) that for any positive integer .
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Paper 4, Section I, E
2013 commentLet be a sequence of real numbers. What does it mean to say that the sequence is convergent? What does it mean to say the series is convergent? Show that if is convergent, then the sequence converges to zero. Show that the converse is not necessarily true.
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Paper 4, Section II, E
2013 comment(i) What does it mean to say that a function is injective? What does it mean to say that is surjective? Let be a function. Show that if is injective, then so is , and that if is surjective, then so is .
(ii) Let be two sets. Their product is the set of ordered pairs with . Let (for be the function
When is surjective? When is injective?
(iii) Now let be any set, and let be functions. Show that there exists a unique such that and .
Show that if or is injective, then is injective. Is the converse true? Justify your answer.
Show that if is surjective then both and are surjective. Is the converse true? Justify your answer.
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Paper 4, Section II, E
2013 comment(i) Let and be integers with . Let be the set of -tuples of non-negative integers satisfying the equation . By mapping elements of to suitable subsets of of size , or otherwise, show that the number of elements of equals
(ii) State the Inclusion-Exclusion principle.
(iii) Let be positive integers. Show that the number of -tuples of integers satisfying
where the binomial coefficient is defined to be zero if .
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Paper 4, Section II,
2013 comment(i) What does it mean to say that a set is countable? Show directly from your definition that any subset of a countable set is countable, and that a countable union of countable sets is countable.
(ii) Let be either or . A function is said to be periodic if there exists a positive integer such that for every . Show that the set of periodic functions from to itself is countable. Is the set of periodic functions countable? Justify your answer.
(iii) Show that is not the union of a countable collection of lines.
[You may assume that and the power set of are uncountable.]
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Paper 4, Section II, E
2013 commentLet be a prime number, and integers with .
(i) Prove Fermat's Little Theorem: for any integer .
(ii) Show that if is an integer such that , then for every integer ,
Deduce that
(iii) Show that there exists a unique integer such that
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Paper 2, Section I, F
2013 commentLet be a random variable with mean and variance . Let
Show that for all . For what value of is there equality?
Let
Supposing that has probability density function , express in terms of . Show that is minimised when is such that .
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Paper 2, Section I, F
2013 comment(i) Let be a random variable. Use Markov's inequality to show that
for all and real .
(ii) Calculate in the case where is a Poisson random variable with parameter . Using the inequality from part (i) with a suitable choice of , prove that
for all .
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Paper 2, Section II, F
2013 commentLet be an exponential random variable with parameter . Show that
for any .
Let be the greatest integer less than or equal to . What is the probability mass function of ? Show that .
Let be the fractional part of . What is the density of ?
Show that and are independent.
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Paper 2, Section II, F
2013 commentLet be a random variable taking values in the non-negative integers, and let be the probability generating function of . Assuming is everywhere finite, show that
where is the mean of and is its variance. [You may interchange differentiation and expectation without justification.]
Consider a branching process where individuals produce independent random numbers of offspring with the same distribution as . Let be the number of individuals in the -th generation, and let be the probability generating function of . Explain carefully why
Assuming , compute the mean of . Show that
Suppose and . Compute the probability that the population will eventually become extinct. You may use standard results on branching processes as long as they are clearly stated.
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Paper 2, Section II, F
2013 commentLet be a geometric random variable with . Derive formulae for and in terms of
A jar contains balls. Initially, all of the balls are red. Every minute, a ball is drawn at random from the jar, and then replaced with a green ball. Let be the number of minutes until the jar contains only green balls. Show that the expected value of is . What is the variance of
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Paper 2, Section II, F
2013 commentLet be the sample space of a probabilistic experiment, and suppose that the sets are a partition of into events of positive probability. Show that
for any event of positive probability.
A drawer contains two coins. One is an unbiased coin, which when tossed, is equally likely to turn up heads or tails. The other is a biased coin, which will turn up heads with probability and tails with probability . One coin is selected (uniformly) at random from the drawer. Two experiments are performed:
(a) The selected coin is tossed times. Given that the coin turns up heads times and tails times, what is the probability that the coin is biased?
(b) The selected coin is tossed repeatedly until it turns up heads times. Given that the coin is tossed times in total, what is the probability that the coin is biased?
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Paper 3, Section I, C
2013 commentThe curve is given by
(i) Compute the arc length of between the points with and .
(ii) Derive an expression for the curvature of as a function of arc length measured from the point with .
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Paper 3, Section , C
2013 commentState a necessary and sufficient condition for a vector field on to be conservative.
Check that the field
is conservative and find a scalar potential for .
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Paper 3, Section II, C
2013 commentGive an explicit formula for which makes the following result hold:
where the region , with coordinates , and the region , with coordinates , are in one-to-one correspondence, and
Explain, in outline, why this result holds.
Let be the region in defined by and . Sketch the region and employ a suitable transformation to evaluate the integral
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Paper 3, Section II, C
2013 commentConsider the bounded surface that is the union of for and for . Sketch the surface.
Using suitable parametrisations for the two parts of , calculate the integral
for .
Check your result using Stokes's Theorem.
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Paper 3, Section II, C
2013 commentIf and are vectors in , show that
is a second rank tensor.
Now assume that and obey Maxwell's equations, which in suitable units read
where is the charge density and the current density. Show that
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Paper 3, Section II, C
2013 comment(a) Prove that
(b) State the divergence theorem for a vector field in a closed region bounded by .
For a smooth vector field and a smooth scalar function prove that
where is the outward unit normal on the surface .
Use this identity to prove that the solution to the Laplace equation in with on is unique, provided it exists.
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Paper 1, Section I,
2013 comment(a) State de Moivre's theorem and use it to derive a formula for the roots of order of a complex number . Using this formula compute the cube roots of .
(b) Consider the equation for . Give a geometric description of the set of solutions and sketch as a subset of the complex plane.
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Paper 1, Section I, A
2013 commentLet be a real matrix.
(i) For with
find an angle so that the element , where denotes the entry of the matrix .
(ii) For with and
show that and find an angle so that .
(iii) For with and
show that and find an angle so that .
(iv) Deduce that any real matrix can be written as a product of an orthogonal matrix and an upper triangular matrix.
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Paper 1, Section II,
2013 commentLet and be non-zero vectors in . What is meant by saying that and are linearly independent? What is the dimension of the subspace of spanned by and if they are (1) linearly independent, (2) linearly dependent?
Define the scalar product for . Define the corresponding norm of . State and prove the Cauchy-Schwarz inequality, and deduce the triangle inequality. Under what condition does equality hold in the Cauchy-Schwarz inequality?
Let be unit vectors in . Let
Show that for any fixed, linearly independent vectors and , the minimum of over is attained when for some , and that for this value of we have
(i) (for any choice of and ;
(ii) and in the case where .
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Paper 1, Section II,
2013 commentDefine the kernel and the image of a linear map from to .
Let be a basis of and a basis of . Explain how to represent by a matrix relative to the given bases.
A second set of bases and is now used to represent by a matrix . Relate the elements of to the elements of .
Let be a linear map from to defined by
Either find one or more in such that
or explain why one cannot be found.
Let be a linear map from to defined by
Find the kernel of .
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Paper 1, Section II, B
2013 comment(a) Let be distinct eigenvalues of an matrix , with corresponding eigenvectors . Prove that the set is linearly independent.
(b) Consider the quadric surface in defined by
Find the position of the origin and orthonormal coordinate basis vectors and , for a coordinate system in which takes the form
Also determine the values of and , and describe the surface geometrically.
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Paper 1, Section II, B
2013 comment(a) Let and be the matrices of a linear map on relative to bases and respectively. In this question you may assume without proof that and are similar.
(i) State how the matrix of relative to the basis is constructed from and . Also state how may be used to compute for any .
(ii) Show that and have the same characteristic equation.
(iii) Show that for any the matrices
are similar. [Hint: if is a basis then so is .]
(b) Using the results of (a), or otherwise, prove that any complex matrix with equal eigenvalues is similar to one of
(c) Consider the matrix
Show that there is a real value such that is an orthogonal matrix. Show that is a rotation and find the axis and angle of the rotation.
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Paper 3, Section I,
2013 commentFor each of the following sequences of functions on , indexed by , determine whether or not the sequence has a pointwise limit, and if so, determine whether or not the convergence to the pointwise limit is uniform.
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Paper 4, Section I,
2013 commentState and prove the chain rule for differentiable mappings and .
Suppose now has image lying on the unit circle in . Prove that the determinant vanishes for every .
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Paper 2, Section I, F
2013 commentLet denote the vector space of continuous real-valued functions on the interval , and let denote the subspace of continuously differentiable functions.
Show that defines a norm on . Show furthermore that the map takes the closed unit ball to a bounded subset of .
If instead we had used the norm restricted from to , would take the closed unit ball to a bounded subset of ? Justify your answer.
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Paper 1, Section II, F
2013 commentDefine what it means for a sequence of functions , to converge uniformly on an interval .
By considering the functions , or otherwise, show that uniform convergence of a sequence of differentiable functions does not imply uniform convergence of their derivatives.
Now suppose is continuously differentiable on for each , that converges as for some , and moreover that the derivatives converge uniformly on . Prove that converges to a continuously differentiable function on , and that
Hence, or otherwise, prove that the function
is continuously differentiable on .
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Paper 4, Section II, F
2013 commentState the contraction mapping theorem.
A metric space is bounded if is a bounded subset of . Suppose is complete and bounded. Let denote the set of continuous from to itself. For , let
Prove that is a complete metric space. Is the subspace of contraction mappings a complete subspace?
Let be the map which associates to any contraction its fixed point. Prove that is continuous.
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Paper 3, Section II, F
2013 commentFor each of the following statements, provide a proof or justify a counterexample.
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The norms and on are Lipschitz equivalent.
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The norms and on the vector space of sequences with are Lipschitz equivalent.
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Given a linear function between normed real vector spaces, there is some for which for every with .
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Given a linear function between normed real vector spaces for which there is some for which for every with , then is continuous.
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The uniform norm is complete on the vector space of continuous real-valued functions on for which for sufficiently large.
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The uniform norm is complete on the vector space of continuous real-valued functions on which are bounded.
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Paper 2, Section II, F
2013 commentLet be continuous on an open set . Suppose that on the partial derivatives and exist and are continuous. Prove that on .
If is infinitely differentiable, and , what is the maximum number of distinct -th order partial derivatives that may have on ?
Let be defined by
Let be defined by
For each of and , determine whether they are (i) differentiable, (ii) infinitely differentiable at the origin. Briefly justify your answers.
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Paper 4, Section I, E
2013 commentState Rouché's theorem. How many roots of the polynomial are contained in the annulus ?
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Paper 1, Section II, E
2013 commentSuppose is a polynomial of even degree, all of whose roots satisfy . Explain why there is a holomorphic (i.e. analytic) function defined on the region which satisfies . We write
By expanding in a Laurent series or otherwise, evaluate
where is the circle of radius 2 with the anticlockwise orientation. (Your answer will be well-defined up to a factor of , depending on which square root you pick.)
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Paper 3, Section II, E
2013 commentLet be the open unit disk, and let be its boundary (the unit circle), with the anticlockwise orientation. Suppose is continuous. Stating clearly any theorems you use, show that
is an analytic function of for .
Now suppose is the restriction of a holomorphic function defined on some annulus . Show that is the restriction of a holomorphic function defined on the open disc .
Let be defined by . Express the coefficients in the power series expansion of centered at 0 in terms of .
Let . What is in the following cases?
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.
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Paper 1, Section I,
2013 commentClassify the singularities (in the finite complex plane) of the following functions: (i) ; (ii) ; (iii) ; (iv) .
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Paper 2, Section II, 13D Let
2013 commentwhere is the rectangle with vertices at and , traversed anti-clockwise.
(i) Show that .
(ii) Assuming that the contribution to from the vertical sides of the rectangle is negligible in the limit , show that
(iii) Justify briefly the assumption that the contribution to from the vertical sides of the rectangle is negligible in the limit .
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Paper 3, Section I, D
2013 commentLet for , and let .
(i) Find the Laplace transforms of and , where is the Heaviside step function.
(ii) Given that the Laplace transform of is , find expressions for the Laplace transforms of and .
(iii) Use Laplace transforms to solve the equation
in the case .
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Paper 4, Section II, D
2013 commentLet and be the circles and , respectively, and let be the (finite) region between the circles. Use the conformal mapping
to solve the following problem:
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Paper 2, Section I, D
2013 commentUse Maxwell's equations to obtain the equation of continuity
Show that, for a body made from material of uniform conductivity , the charge density at any fixed internal point decays exponentially in time. If the body is finite and isolated, explain how this result can be consistent with overall charge conservation.
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Paper 4, Section I, D
2013 commentThe infinite plane is earthed and the infinite plane carries a charge of per unit area. Find the electrostatic potential between the planes.
Show that the electrostatic energy per unit area (of the planes constant) between the planes can be written as either or , where is the potential at .
The distance between the planes is now increased by , where is small. Show that the change in the energy per unit area is if the upper plane is electrically isolated, and is approximately if instead the potential on the upper plane is maintained at . Explain briefly how this difference can be accounted for.
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Paper 1, Section II,
2013 commentBriefly explain the main assumptions leading to Drude's theory of conductivity. Show that these assumptions lead to the following equation for the average drift velocity of the conducting electrons:
where and are the mass and charge of each conducting electron, is the probability that a given electron collides with an ion in unit time, and is the applied electric field.
Given that and , where and are independent of , show that
Here, and is the number of conducting electrons per unit volume.
Now let and , where and are constant. Assuming that remains valid, use Maxwell's equations (taking the charge density to be everywhere zero but allowing for a non-zero current density) to show that
where the relative permittivity and .
In the case and , where , show that the wave decays exponentially with distance inside the conductor.
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Paper 3, Section II, D
2013 commentThree sides of a closed rectangular circuit are fixed and one is moving. The circuit lies in the plane and the sides are , where is a given function of time. A magnetic field is applied, where is a given function of and only. Find the magnetic flux of through the surface bounded by .
Find an electric field that satisfies the Maxwell equation
and then write down the most general solution in terms of and an undetermined scalar function independent of .
Verify that
where is the velocity of the relevant side of . Interpret the left hand side of this equation.
If a unit current flows round , what is the rate of work required to maintain the motion of the moving side of the rectangle? You should ignore any electromagnetic fields produced by the current.
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Paper 2, Section II, D
2013 commentStarting with the expression
for the magnetic vector potential at the point due to a current distribution of density , obtain the Biot-Savart law for the magnetic field due to a current flowing in a simple loop :
Verify by direct differentiation that this satisfies . You may use without proof the identity , where is a constant vector and is a vector field.
Given that is planar, and is described in cylindrical polar coordinates by , , show that the magnetic field at the origin is
If is the ellipse , find the magnetic field at the focus due to a current .
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Paper 1, Section I, A
2013 commentA two-dimensional flow is given by
Show that the flow is both irrotational and incompressible. Find a stream function such that . Sketch the streamlines at .
Find the pathline of a fluid particle that passes through at in the form and sketch the pathline for
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Paper 2, Section I, A
2013 commentAn incompressible, inviscid fluid occupies the region beneath the free surface and moves with a velocity field determined by the velocity potential Gravity acts in the direction. You may assume Bernoulli's integral of the equation of motion:
Give the kinematic and dynamic boundary conditions that must be satisfied by on .
In the absence of waves, the fluid has constant uniform velocity in the direction. Derive the linearised form of the boundary conditions for small amplitude waves.
Assume that the free surface and velocity potential are of the form:
(where implicitly the real parts are taken). Show that
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Paper 1, Section II, A
2013 commentStarting from the Euler momentum equation, derive the form of Bernoulli's equation appropriate for an unsteady irrotational motion of an inviscid incompressible fluid.
Water of density is driven through a horizontal tube of length and internal radius from a water-filled balloon attached to one end of the tube. Assume that the pressure exerted by the balloon is proportional to its current volume (in excess of atmospheric pressure). Also assume that water exits the tube at atmospheric pressure, and that gravity may be neglected. Show that the time for the balloon to empty does not depend on its initial volume. Find the maximum speed of water exiting the pipe.
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Paper 4, Section II, A
2013 commentThe axisymmetric, irrotational flow generated by a solid sphere of radius translating at velocity in an inviscid, incompressible fluid is represented by a velocity potential . Assume the fluid is at rest far away from the sphere. Explain briefly why .
By trying a solution of the form , show that
and write down the fluid velocity.
Show that the total kinetic energy of the fluid is where is the mass of the sphere and is the ratio of the density of the fluid to the density of the sphere.
A heavy sphere (i.e. ) is released from rest in an inviscid fluid. Determine its speed after it has fallen a distance in terms of and .
Note, in spherical polars:
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Paper 3, Section II, A
2013 commentA layer of incompressible fluid of density and viscosity flows steadily down a plane inclined at an angle to the horizontal. The layer is of uniform thickness measured perpendicular to the plane and the viscosity of the overlying air can be neglected. Using coordinates parallel to the plane (in steepest downwards direction) and normal to the plane, write down the equations of motion and the boundary conditions on the plane and on the free top surface. Determine the pressure and velocity fields and show that the volume flux down the plane is
Consider now the case where a second layer of fluid, of uniform thickness , viscosity and density , flows steadily on top of the first layer. Explain why one of the appropriate boundary conditions between the two fluids is
where is the component of velocity in the direction and and refer to just below and just above the boundary respectively. Determine the velocity field in each layer.
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Paper 1, Section I, F
2013 commentLet and be ultraparallel geodesics in the hyperbolic plane. Prove that the have a unique common perpendicular.
Suppose now are pairwise ultraparallel geodesics in the hyperbolic plane. Can the three common perpendiculars be pairwise disjoint? Must they be pairwise disjoint? Briefly justify your answers.
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Paper 3, Section I, F
2013 commentLet be a surface with Riemannian metric having first fundamental form . State a formula for the Gauss curvature of .
Suppose that is flat, so vanishes identically, and that is a geodesic on when parametrised by arc-length. Using the geodesic equations, or otherwise, prove that , i.e. is locally isometric to a plane.
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Paper 2, Section II, F
2013 commentLet and be disjoint circles in . Prove that there is a Möbius transformation which takes and to two concentric circles.
A collection of circles , for which
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is tangent to and , where indices are ;
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the circles are disjoint away from tangency points;
is called a constellation on . Prove that for any there is some pair and a constellation on made up of precisely circles. Draw a picture illustrating your answer.
Given a constellation on , prove that the tangency points for all lie on a circle. Moreover, prove that if we take any other circle tangent to and , and then construct for inductively so that is tangent to and , then we will have , i.e. the chain of circles will again close up to form a constellation.
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Paper 3, Section II, F
2013 commentShow that the set of all straight lines in admits the structure of an abstract smooth surface . Show that is an open Möbius band (i.e. the Möbius band without its boundary circle), and deduce that admits a Riemannian metric with vanishing Gauss curvature.
Show that there is no metric , in the sense of metric spaces, which
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induces the locally Euclidean topology on constructed above;
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is invariant under the natural action on of the group of translations of .
Show that the set of great circles on the two-dimensional sphere admits the structure of a smooth surface . Is homeomorphic to ? Does admit a Riemannian metric with vanishing Gauss curvature? Briefly justify your answers.
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Paper 4, Section II, F
2013 commentLet be a smooth curve in the -plane , with for every and . Let be the surface obtained by rotating around the -axis. Find the first fundamental form of .
State the equations for a curve parametrised by arc-length to be a geodesic.
A parallel on is the closed circle swept out by rotating a single point of . Prove that for every there is some for which exactly parallels are geodesics. Sketch possible such surfaces in the cases and .
If every parallel is a geodesic, what can you deduce about ? Briefly justify your answer.
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Paper 3, Section I,
2013 commentDefine the notion of a free module over a ring. When is a PID, show that every ideal of is free as an -module.
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Paper 4, Section I,
2013 commentLet be a prime number, and be a non-trivial finite group whose order is a power of . Show that the size of every conjugacy class in is a power of . Deduce that the centre of has order at least .
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Paper 2, Section I, G
2013 commentShow that every Euclidean domain is a PID. Define the notion of a Noetherian ring, and show that is Noetherian by using the fact that it is a Euclidean domain.
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Paper 1, Section II, G
2013 comment(i) Consider the group of all 2 by 2 matrices with entries in and non-zero determinant. Let be its subgroup consisting of all diagonal matrices, and be the normaliser of in . Show that is generated by and , and determine the quotient group .
(ii) Now let be a prime number, and be the field of integers modulo . Consider the group as above but with entries in , and define and similarly. Find the order of the group .
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Paper 4, Section II, 11G
2013 commentLet be an integral domain, and be a finitely generated -module.
(i) Let be a finite subset of which generates as an -module. Let be a maximal linearly independent subset of , and let be the -submodule of generated by . Show that there exists a non-zero such that for every .
(ii) Now assume is torsion-free, i.e. for and implies or . By considering the map mapping to for as in (i), show that every torsion-free finitely generated -module is isomorphic to an -submodule of a finitely generated free -module.
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Paper 3, Section II, G
2013 commentLet be the polynomial ring in two variables over the complex numbers, and consider the principal ideal of .
(i) Using the fact that is a UFD, show that is a prime ideal of . [Hint: Elements in are polynomials in with coefficients in
(ii) Show that is not a maximal ideal of , and that it is contained in infinitely many distinct proper ideals in .
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Paper 2, Section II, G
2013 comment(i) State the structure theorem for finitely generated modules over Euclidean domains.
(ii) Let be the polynomial ring over the complex numbers. Let be a module which is 4-dimensional as a -vector space and such that for all . Find all possible forms we obtain when we write for irreducible and .
(iii) Consider the quotient ring as a -module. Show that is isomorphic as a -module to the direct sum of three copies of . Give the isomorphism and its inverse explicitly.
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Paper 4, Section I, E
2013 commentWhat is a quadratic form on a finite dimensional real vector space ? What does it mean for two quadratic forms to be isomorphic (i.e. congruent)? State Sylvester's law of inertia and explain the definition of the quantities which appear in it. Find the signature of the quadratic form on given by , where
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Paper 2, Section I, E
2013 commentIf is an invertible Hermitian matrix, let
Show that with the operation of matrix multiplication is a group, and that det has norm 1 for any . What is the relation between and the complex Hermitian form defined by ?
If is the identity matrix, show that any element of is diagonalizable.
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Paper 1, Section I, E
2013 commentWhat is the adjugate of an matrix ? How is it related to ? Suppose all the entries of are integers. Show that all the entries of are integers if and only if .
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Paper 1, Section II, E
2013 commentIf and are vector spaces, what is meant by ? If and are subspaces of a vector space , what is meant by ?
Stating clearly any theorems you use, show that if and are subspaces of a finite dimensional vector space , then
Let be subspaces with bases
Find a basis for such that the first component of and the second component of are both 0 .
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Paper 4, Section II, E
2013 commentWhat does it mean for an matrix to be in Jordan form? Show that if is in Jordan form, there is a sequence of diagonalizable matrices which converges to , in the sense that the th component of converges to the th component of for all and . [Hint: A matrix with distinct eigenvalues is diagonalizable.] Deduce that the same statement holds for all .
Let . Given , define a linear map by . Express the characteristic polynomial of in terms of the trace and determinant of . [Hint: First consider the case where is diagonalizable.]
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Paper 3, Section II, E
2013 commentLet and be finite dimensional real vector spaces and let be a linear map. Define the dual space and the dual map . Show that there is an isomorphism which is canonical, in the sense that for any automorphism of .
Now let be an inner product space. Use the inner product to show that there is an injective map from im to . Deduce that the row rank of a matrix is equal to its column rank.
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Paper 2, Section II, E
2013 commentDefine what it means for a set of vectors in a vector space to be linearly dependent. Prove from the definition that any set of vectors in is linearly dependent.
Using this or otherwise, prove that if has a finite basis consisting of elements, then any basis of has exactly elements.
Let be the vector space of bounded continuous functions on . Show that is infinite dimensional.
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Paper 4, Section I, H
2013 commentSuppose is the transition matrix of an irreducible recurrent Markov chain with state space . Show that if is an invariant measure and for some , then for all .
Let
Give a meaning to and explain why .
Suppose is an invariant measure with . Prove that for all .
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Paper 3, Section I, H
2013 commentProve that if a distribution is in detailed balance with a transition matrix then it is an invariant distribution for .
Consider the following model with 2 urns. At each time, one of the following happens:
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with probability a ball is chosen at random and moved to the other urn (but nothing happens if both urns are empty);
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with probability a ball is chosen at random and removed (but nothing happens if both urns are empty);
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with probability a new ball is added to a randomly chosen urn,
where and . State denotes that urns 1,2 contain and balls respectively. Prove that there is an invariant measure
Find the proportion of time for which there are balls in the system.
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Paper 1, Section II, 20H
2013 commentA Markov chain has state space and transition matrix
where the rows correspond to , respectively. Show that this Markov chain is equivalent to a random walk on some graph with 6 edges.
Let denote the mean first passage time from to .
(i) Find and .
(ii) Given , find the expected number of steps until the walk first completes a step from to .
(iii) Suppose the distribution of is . Let be the least such that appears as a subsequence of . By comparing the distributions of and show that and that
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Paper 2, Section II, H
2013 comment(i) Suppose is an irreducible Markov chain and for some . Prove that and that
(ii) Let be a symmetric random walk on the lattice. Prove that is recurrent. You may assume, for ,
(iii) A princess and monster perform independent random walks on the lattice. The trajectory of the princess is the symmetric random walk . The monster's trajectory, denoted , is a sleepy version of an independent symmetric random walk . Specifically, given an infinite sequence of integers , the monster sleeps between these times, so . Initially, and . The princess is captured if and only if at some future time she and the monster are simultaneously at .
Compare the capture probabilities for an active monster, who takes for all , and a sleepy monster, who takes spaced sufficiently widely so that
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Paper 2, Section I, B
2013 commentConsider the equation
subject to the Cauchy data . Using the method of characteristics, obtain a solution to this equation.
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Paper 4, Section I, C
2013 commentShow that the general solution of the wave equation
can be written in the form
For the boundary conditions
find the relation between and and show that they are -periodic. Hence show that
is independent of .
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Paper 3, Section I, C
2013 commentThe solution to the Dirichlet problem on the half-space :
is given by the formula
where is the outward normal to .
State the boundary conditions on and explain how is related to , where
is the fundamental solution to the Laplace equation in three dimensions.
Using the method of images find an explicit expression for the function in the formula.
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Paper 1, Section II, B
2013 comment(i) Let . Obtain the Fourier sine series and sketch the odd and even periodic extensions of over the interval . Deduce that
(ii) Consider the eigenvalue problem
with boundary conditions . Find the eigenvalues and corresponding eigenfunctions. Recast in Sturm-Liouville form and give the orthogonality condition for the eigenfunctions. Using the Fourier sine series obtained in part (i), or otherwise, and assuming completeness of the eigenfunctions, find a series for that satisfies
for the given boundary conditions.
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Paper 3, Section II, C
2013 commentThe Laplace equation in plane polar coordinates has the form
Using separation of variables, derive the general solution to the equation that is singlevalued in the domain .
For
solve the Laplace equation in the annulus with the boundary conditions:
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Paper 2, Section II, B
2013 commentThe steady-state temperature distribution in a uniform rod of finite length satisfies the boundary value problem
where is the (constant) diffusion coefficient. Determine the Green's function for this problem. Now replace the above homogeneous boundary conditions with the inhomogeneous boundary conditions and give a solution to the new boundary value problem. Hence, obtain the steady-state solution for the following problem with the specified boundary conditions:
[You may assume that a steady-state solution exists.]
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Paper 4, Section II, C
2013 commentFind the inverse Fourier transform of the function
Assuming that appropriate Fourier transforms exist, determine the solution of
with the following boundary conditions
Here is the Dirac delta-function.
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Paper 3, Section I, G
2013 commentLet be a metric space with the metric .
(i) Show that if is compact as a topological space, then is complete.
(ii) Show that the completeness of is not a topological property, i.e. give an example of two metrics on a set , such that the associated topologies are the same, but is complete and is not.
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Paper 2, Section I, G
2013 commentLet be a topological space. Prove or disprove the following statements.
(i) If is discrete, then is compact if and only if it is a finite set.
(ii) If is a subspace of and are both compact, then is closed in .
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Paper 1, Section II, G
2013 commentConsider the sphere , a subset of , as a subspace of with the Euclidean metric.
(i) Show that is compact and Hausdorff as a topological space.
(ii) Let be the quotient set with respect to the equivalence relation identifying the antipodes, i.e.
Show that is compact and Hausdorff with respect to the quotient topology.
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Paper 4, Section II, G
2013 commentLet be a topological space. A connected component of means an equivalence class with respect to the equivalence relation on defined as:
(i) Show that every connected component is a connected and closed subset of .
(ii) If are topological spaces and is the product space, show that every connected component of is a direct product of connected components of and .
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Paper 1, Section I, C
2013 commentDetermine the nodes of the two-point Gaussian quadrature
and express the coefficients in terms of . [You don't need to find numerical values of the coefficients.]
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Paper 4, Section I, C
2013 commentFor a continuous function , and distinct points , define the divided difference of order .
Given points , let be the polynomial of degree that interpolates at these points. Prove that can be written in the Newton form
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Paper 1, Section II, C
2013 commentDefine the QR factorization of an matrix and explain how it can be used to solve the least squares problem of finding the vector which minimises , where , and the norm is the Euclidean one.
Define a Givens rotation and show that it is an orthogonal matrix.
Using a Givens rotation, solve the least squares problem for
giving both and .
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Paper 3, Section II, C
2013 commentbe a formula of numerical differentiation which is exact on polynomials of degree 2 , and let
be its error.
Find the values of the coefficients .
Using the Peano kernel theorem, find the least constant such that, for all functions , we have
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Paper 2, Section II, C
2013 commentExplain briefly what is meant by the convergence of a numerical method for solving the ordinary differential equation
Prove from first principles that if the function is sufficiently smooth and satisfies the Lipschitz condition
for some , then the backward Euler method
converges and find the order of convergence.
Find the linear stability domain of the backward Euler method.
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Paper 1, Section I,
2013 commentState sufficient conditions for and to be optimal mixed strategies for the row and column players in a zero-sum game with payoff matrix and value .
Rowena and Colin play a hide-and-seek game. Rowena hides in one of 3 locations, and then Colin searches them in some order. If he searches in order then his search cost is or , depending upon whether Rowena hides in or , respectively, and where are all positive. Rowena (Colin) wishes to maximize (minimize) the expected search cost.
Formulate the payoff matrix for this game.
Let . Suppose that Colin starts his search in location with probability , and then, if he does not find Rowena, he searches the remaining two locations in random order. What bound does this strategy place on the value of the game?
Guess Rowena's optimal hiding strategy, show that it is optimal and find the value of the game.
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Paper 2, Section I, H
2013 commentGiven a network with a source , a sink , and capacities on directed arcs, define what is meant by a minimum cut.
The streets and intersections of a town are represented by sets of edges and vertices of a connected graph. A city planner wishes to make all streets one-way while ensuring it possible to drive away from each intersection along at least different streets.
Use a theorem about min-cut and max-flow to prove that the city planner can achieve his goal provided that the following is true:
where is the size of and is the number edges with at least one end in . How could the planner find street directions that achieve his goal?
[Hint: Consider a network having nodes , nodes for the streets and nodes for the intersections. There are directed arcs from to each and from each to . From each there are two further arcs, directed towards and that correspond to endpoints of street
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Paper 4, Section II, 20H
2013 commentGiven real numbers and , consider the problem of minimizing
subject to and
List all the basic feasible solutions, writing them as matrices .
Let . Suppose there exist such that
Prove that if and are both feasible for and whenever , then
Let be the initial feasible solution that is obtained by formulating as a transportation problem and using a greedy method that starts in the upper left of the matrix . Show that if then minimizes .
For what values of and is one step of the transportation algorithm sufficient to pivot from to a solution that maximizes ?
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Paper 3, Section II, H
2013 commentUse the two phase method to find all optimal solutions to the problem
Suppose that the values are perturbed to . Find an expression for the change in the optimal value, which is valid for all sufficiently small values of .
Suppose that . For what values of is your expression valid?
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Paper 4, Section I, B
2013 commentThe components of the three-dimensional angular momentum operator are defined as follows:
Given that the wavefunction
is an eigenfunction of , find all possible values of and the corresponding eigenvalues of . Letting , show that is an eigenfunction of and calculate the corresponding eigenvalue.
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Paper 3, Section I, B
2013 commentIf and are linear operators, establish the identity
In what follows, the operators and are Hermitian and represent position and momentum of a quantum mechanical particle in one-dimension. Show that
and
where . Assuming , show that the operators and are Hermitian but their product is not. Determine whether is Hermitian.
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Paper 1, Section II, B
2013 commentA particle with momentum moves in a one-dimensional real potential with Hamiltonian given by
where is a real function and . Obtain the potential energy of the system. Find such that . Now, putting , for , show that can be normalised only if is odd. Letting , use the inequality
to show that
assuming that both and vanish.
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Paper 3, Section II, B
2013 commentObtain, with the aid of the time-dependent Schrödinger equation, the conservation equation
where is the probability density and is the probability current. What have you assumed about the potential energy of the system?
Show that if the potential is complex the conservation equation becomes
Take the potential to be time-independent. Show, with the aid of the divergence theorem, that
Assuming the wavefunction is normalised to unity, show that if is expanded about so that , then
As time increases, how does the quantity on the left of this equation behave if ?
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Paper 2, Section II, B
2013 comment(i) Consider a particle of mass confined to a one-dimensional potential well of depth and potential
If the particle has energy where , show that for even states
where and .
(ii) A particle of mass that is incident from the left scatters off a one-dimensional potential given by
where is the Dirac delta. If the particle has energy and , obtain the reflection and transmission coefficients and , respectively. Confirm that .
For the case and show that the energy of the only even parity bound state of the system is given by
Use part (i) to verify this result by taking the limit with fixed.
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Paper 1, Section I, H
2013 commentLet be independent and identically distributed observations from a distribution with probability density function
where and are unknown positive parameters. Let . Find the maximum likelihood estimators and .
Determine for each of and whether or not it has a positive bias.
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Paper 2, Section I, H
2013 commentState and prove the Rao-Blackwell theorem.
Individuals in a population are independently of three types , with unknown probabilities where . In a random sample of people the th person is found to be of type .
Show that an unbiased estimator of is
Suppose that of the individuals are of type . Find an unbiased estimator of , say , such that .
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Paper 4, Section II, H
2013 commentExplain the notion of a sufficient statistic.
Suppose is a random variable with distribution taking values in , with . Let be a sample from . Suppose is the number of these that are equal to . Use a factorization criterion to explain why is sufficient for .
Let be the hypothesis that for all . Derive the statistic of the generalized likelihood ratio test of against the alternative that this is not a good fit.
Assuming that when is true and is large, show that this test can be approximated by a chi-squared test using a test statistic
Suppose and . Would you reject Explain your answer.
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Paper 1, Section II, H
2013 commentConsider the general linear model where is a known matrix, is an unknown vector of parameters, and is an vector of independent random variables with unknown variance . Assume the matrix is invertible. Let
What are the distributions of and ? Show that and are uncorrelated.
Four apple trees stand in a rectangular grid. The annual yield of the tree at coordinate conforms to the model
where is the amount of fertilizer applied to tree may differ because of varying soil across rows, and the are random variables that are independent of one another and from year to year. The following two possible experiments are to be compared:
Represent these as general linear models, with . Compare the variances of estimates of under I and II.
With II the following yields are observed:
Forecast the total yield that will be obtained next year if no fertilizer is used. What is the predictive interval for this yield?
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Paper 3, Section II, H
2013 commentSuppose is a single observation from a distribution with density over . It is desired to test against .
Let define a test by 'accept '. Let . State the Neyman-Pearson lemma using this notation.
Let be the best test of size . Find and .
Consider now where means 'declare the test to be inconclusive'. Let . Given prior probabilities for and for , and some , let
Let , where . Prove that for each value of there exist (depending on such that Hint
Hence prove that if is any test for which
then and .
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Paper 1, Section I, A
2013 comment(a) Define what it means for a function to be convex. Assuming exists, state an equivalent condition. Let , defined on . Show that is convex.
(b) Find the Legendre transform of . State the domain of . Without further calculation, explain why in this case.
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Paper 3, Section II, H
2013 commentLet be the plane curve given by the polynomial
over the field of complex numbers, where .
(i) Show that is nonsingular.
(ii) Compute the divisors of the rational functions
on .
(iii) Consider the morphism . Compute its ramification points and degree.
(iv) Show that a basis for the space of regular differentials on is
where
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Paper 4, Section II, H
2013 commentLet be a nonsingular projective curve, and a divisor on of degree .
(i) State the Riemann-Roch theorem for , giving a brief explanation of each term. Deduce that if then .
(ii) Show that, for every ,
Deduce that . Show also that if , then for all but finitely many .
(iii) Deduce that for every there exists a divisor of degree with .
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Paper 2, Section II, H
2013 commentLet be an irreducible quadric surface.
(i) Show that if is singular, then every nonsingular point lies in exactly one line in , and that all the lines meet in the singular point, which is unique.
(ii) Show that if is nonsingular then each point of lies on exactly two lines of .
Let be nonsingular, a point of , and a plane not containing . Show that the projection from to is a birational map . At what points does fail to be regular? At what points does fail to be regular? Justify your answers.
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Paper 1, Section II, H
2013 commentLet be an affine variety over an algebraically closed field . What does it mean to say that is irreducible? Show that any non-empty affine variety is the union of a finite number of irreducible affine varieties .
Define the ideal of . Show that is a prime ideal if and only if is irreducible.
Assume that the base field has characteristic zero. Determine the irreducible components of
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Paper 3, Section II, G
2013 comment(i) State, but do not prove, the Mayer-Vietoris theorem for the homology groups of polyhedra.
(ii) Calculate the homology groups of the -sphere, for every .
(iii) Suppose that and . Calculate the homology groups of the subspace of defined by .
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Paper 4, Section II, G
2013 comment(i) State, but do not prove, the Lefschetz fixed point theorem.
(ii) Show that if is even, then for every map there is a point such that . Is this true if is odd? [Standard results on the homology groups for the -sphere may be assumed without proof, provided they are stated clearly.]
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Paper 2, Section II, G
2013 comment(i) State the Seifert-van Kampen theorem.
(ii) Assuming any standard results about the fundamental group of a circle that you wish, calculate the fundamental group of the -sphere, for every .
(iii) Suppose that and that is a path-connected topological -manifold. Show that is isomorphic to for any .
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Paper 1, Section II, 21G
2013 comment(i) Define the notion of the fundamental group of a path-connected space with base point .
(ii) Prove that if a group acts freely and properly discontinuously on a simply connected space , then is isomorphic to . [You may assume the homotopy lifting property, provided that you state it clearly.]
(iii) Suppose that are distinct points on the 2 -sphere and that . Exhibit a simply connected space with an action of a group as in (ii) such that , and calculate .
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Paper 4, Section II, D
2013 commentDefine the Floquet matrix for a particle moving in a periodic potential in one dimension and explain how it determines the allowed energy bands of the system.
A potential barrier in one dimension has the form
where is a smooth, positive function of . The reflection and transmission amplitudes for a particle of wavenumber , incident from the left, are and respectively. For a particle of wavenumber , incident from the right, the corresponding amplitudes are and . In the following, for brevity, we will suppress the -dependence of these quantities.
Consider the periodic potential , defined by for and by elsewhere. Write down two linearly independent solutions of the corresponding Schrödinger equation in the region . Using the scattering data given above, extend these solutions to the region . Hence find the Floquet matrix of the system in terms of the amplitudes and defined above.
Show that the edges of the allowed energy bands for this potential lie at , where
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Paper 2, Section II, D
2013 comment(i) A particle of momentum and energy scatters off a sphericallysymmetric target in three dimensions. Define the corresponding scattering amplitude as a function of the scattering angle . Expand the scattering amplitude in partial waves of definite angular momentum , and determine the coefficients of this expansion in terms of the phase shifts appearing in the following asymptotic form of the wavefunction, valid at large distance from the target,
Here, is the distance from the target and are the Legendre polynomials.
[You may use without derivation the following approximate relation between plane and spherical waves (valid asymptotically for large ):
(ii) Suppose that the potential energy takes the form where is a dimensionless coupling. By expanding the wavefunction in a power series in , derive the Born Approximation to the scattering amplitude in the form
up to corrections of order , where . [You may quote any results you need for the Green's function for the differential operator provided they are stated clearly.]
(iii) Derive the corresponding order contribution to the phase shift of angular momentum .
[You may use the orthogonality relations
and the integral formula
where is a spherical Bessel function.]
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Paper 3, Section II, D
2013 commentWrite down the classical Hamiltonian for a particle of mass , electric charge and momentum p moving in the background of an electromagnetic field with vector and scalar potentials and .
Consider the case of a constant uniform magnetic field, and . Working in the gauge with and , show that Hamilton's equations,
admit solutions corresponding to circular motion in the plane with angular frequency .
Show that, in the same gauge, the coordinates of the centre of the circle are related to the instantaneous position and momentum of the particle by
Write down the quantum Hamiltonian for the system. In the case of a uniform constant magnetic field discussed above, find the allowed energy levels. Working in the gauge specified above, write down quantum operators corresponding to the classical quantities and defined in (1) above and show that they are conserved.
[In this question you may use without derivation any facts relating to the energy spectrum of the quantum harmonic oscillator provided they are stated clearly.]
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Paper 1, Section II, D
2013 commentConsider a quantum system with Hamiltonian and energy levels
For any state define the Rayleigh-Ritz quotient and show the following:
(i) the ground state energy is the minimum value of ;
(ii) all energy eigenstates are stationary points of with respect to variations of .
Under what conditions can the value of for a trial wavefunction (depending on some parameter ) be used as an estimate of the energy of the first excited state? Explain your answer.
For a suitably chosen trial wavefunction which is the product of a polynomial and a Gaussian, use the Rayleigh-Ritz quotient to estimate for a particle of mass moving in a potential , where is a constant.
[You may use the integral formulae,
where is a non-negative integer and is a constant. ]
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Paper 4, Section II, J
2013 comment(i) Define an queue. Justifying briefly your answer, specify when this queue has a stationary distribution, and identify that distribution. State and prove Burke's theorem for this queue.
(ii) Let denote a Jackson network of queues, where the entrance and service rates for queue are respectively and , and each customer leaving queue moves to queue with probability after service. We assume for each ; with probability a customer leaving queue departs from the system. State Jackson's theorem for this network. [You are not required to prove it.] Are the processes independent at equilibrium? Justify your answer.
(iii) Let be the process of final departures from queue . Show that, at equilibrium, is independent of . Show that, for each fixed is a Poisson process, and specify its rate.
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Paper 3, Section II, J
2013 commentDefine the Moran model. Describe briefly the infinite sites model of mutations.
We henceforth consider a population with individuals evolving according to the rules of the Moran model. In addition we assume:
-
the allelic type of any individual at any time lies in a given countable state space ;
-
individuals are subject to mutations at constant rate , independently of the population dynamics;
-
each time a mutation occurs, if the allelic type of the individual was , it changes to with probability , where is a given Markovian transition matrix on that is symmetric:
(i) Show that, if two individuals are sampled at random from the population at some time , then the time to their most recent common ancestor has an exponential distribution, with a parameter that you should specify.
(ii) Let be the total number of mutations that accumulate on the two branches separating these individuals from their most recent common ancestor. Show that is a geometric random variable, and specify its probability parameter .
(iii) The first individual is observed to be of type . Explain why the probability that the second individual is also of type is
where is a Markov chain on with transition matrix and is independent of .
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Paper 2, Section II, J
2013 comment(i) Define a Poisson process as a Markov chain on the non-negative integers and state three other characterisations.
(ii) Let be a continuous positive function. Let be a right-continuous process with independent increments, such that
where the terms are uniform in . Show that is a Poisson random variable with parameter .
(iii) Let be a sequence of independent and identically distributed positive random variables with continuous density function . We define the sequence of successive records, , by and, for ,
The record process,, is then defined by
Explain why the increments of are independent. Show that is a Poisson random variable with parameter where .
[You may assume the following without proof: For fixed , let (respectively, ) be the subsequence of obtained by retaining only those elements that are greater than (respectively, smaller than) . Then (respectively, ) is a sequence of independent variables each having the distribution of conditioned on (respectively, ); and and are independent.]
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Paper 1, Section II, J
2013 commentLet be a Markov chain on with -matrix given by
where .
(i) Show that is transient if and only if . [You may assume without proof that for all and all sufficiently small positive .]
(ii) Assume that . Find a necessary and sufficient condition for to be almost surely explosive. [You may assume without proof standard results about pure birth processes, provided that they are stated clearly.]
(iii) Find a stationary measure for . For the case and , show that is positive recurrent if and only if .
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Paper 4, Section II, B
2013 commentShow that the equation
has an irregular singular point at infinity. Using the Liouville-Green method, show that one solution has the asymptotic expansion
as
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Paper 3, Section II, B
2013 commentLet
where and are smooth, and for also , , and . Show that, as ,
Consider the Bessel function
Show that, as ,
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Paper 1, Section II, B
2013 commentSuppose . Define what it means to say that
is an asymptotic expansion of as . Show that has no other asymptotic expansion in inverse powers of as .
To estimate the value of for large , one may use an optimal truncation of the asymptotic expansion. Explain what is meant by this, and show that the error is an exponentially small quantity in .
Derive an integral respresentation for a function with the above asymptotic expansion.
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Paper 4, Section I, B
2013 commentThe Lagrangian for a heavy symmetric top of mass , pinned at point which is a distance from the centre of mass, is
(i) Starting with the fixed space frame and choosing at its origin, sketch the top with embedded body frame axis being the symmetry axis. Clearly identify the Euler angles .
(ii) Obtain the momenta and and the Hamiltonian . Derive Hamilton's equations. Identify the three conserved quantities.
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Paper 3, Section I, B
2013 commentTwo equal masses are connected to each other and to fixed points by three springs of force constant and as shown in the figure.

(i) Write down the Lagrangian and derive the equations describing the motion of the system in the direction parallel to the springs.
(ii) Find the normal modes and their frequencies. Comment on your results.
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Paper 2, Section I, B
2013 comment(i) Consider a rigid body with principal moments of inertia . Derive Euler's equations of torque-free motion,
with components of the angular velocity given in the body frame.
(ii) Use Euler's equations to show that the energy and the square of the total angular momentum of the body are conserved.
(iii) Consider a torque-free motion of a symmetric top with . Show that in the body frame the vector of angular velocity precesses about the body-fixed axis with constant angular frequency equal to .
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Paper 1, Section I, B
2013 commentConsider an -dimensional dynamical system with generalized coordinates and momenta .
(a) Define the Poisson bracket of two functions and .
(b) Assuming Hamilton's equations of motion, prove that if a function Poisson commutes with the Hamiltonian, that is , then is a constant of the motion.
(c) Assume that is an ignorable coordinate, that is the Hamiltonian does not depend on it explicitly. Using the formalism of Poisson brackets prove that the conjugate momentum is conserved.
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Paper 4, Section II, B
2013 commentThe motion of a particle of charge and mass in an electromagnetic field with scalar potential and vector potential is characterized by the Lagrangian
(i) Write down the Hamiltonian of the particle.
(ii) Write down Hamilton's equations of motion for the particle.
(iii) Show that Hamilton's equations are invariant under the gauge transformation
for an arbitrary function .
(iv) The particle moves in the presence of a field such that and , where are Cartesian coordinates and is a constant.
(a) Find a gauge transformation such that only one component of remains non-zero.
(b) Determine the motion of the particle.
(v) Now assume that varies very slowly with time on a time-scale much longer than . Find the quantity which remains approximately constant throughout the motion.
[You may use the expression for the action variable .]
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Paper 2, Section II, B
2013 comment(i) The action for a system with a generalized coordinate is given by
(a) State the Principle of Least Action and derive the Euler-Lagrange equation.
(b) Consider an arbitrary function . Show that leads to the same equation of motion.
(ii) A wire frame in a shape of an equilateral triangle with side rotates in a horizontal plane with constant angular frequency about a vertical axis through . A bead of mass is threaded on and moves without friction. The bead is connected to and by two identical light springs of force constant and equilibrium length .
(a) Introducing the displacement of the particle from the mid point of , determine the Lagrangian .
(b) Derive the equation of motion. Identify the integral of the motion.
(c) Describe the motion of the bead. Find the condition for there to be a stable equilibrium and find the frequency of small oscillations about it when it exists.
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Paper 4, Section I,
2013 commentDescribe how a stream cipher works. What is a one-time pad?
A one-time pad is used to send the message which is encoded as 0101011. In error, it is reused to send the message which is encoded as 0100010 . Show that there are two possibilities for the substring , and find them.
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Paper 3, Section I, H
2013 commentDescribe briefly the Rabin cipher with modulus , explaining how it can be deciphered by the intended recipient and why it is difficult for an eavesdropper to decipher it.
The Cabinet decides to communicate using Rabin ciphers to maintain confidentiality. The Cabinet Secretary encrypts a message, represented as a positive integer , using the Rabin cipher with modulus (with ) and publishes both the encrypted message and the modulus. The Defence Secretary deciphers this message to read it but then foolishly encrypts it again using a Rabin cipher with a different modulus (with and publishes the newly encrypted message and . Mr Rime (the Leader of the Opposition) knows this has happened. Explain how Rime can work out what the original message was using the two different encrypted versions.
Can Rime decipher other messages sent out by the Cabinet using the original modulus ?
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Paper 2, Section I, H
2013 commentLet denote the maximum size of a binary code of length with minimum distance . For fixed with , let . Show that
where .
[You may assume the GSV and Hamming bounds and any form of Stirling's theorem provided you state them clearly.]
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Paper 1, Section I, H
2013 commentA binary Huffman code is used for encoding symbols occurring with respective probabilities where . Let be the length of a shortest codeword and the length of a longest codeword. Determine the maximal and minimal values of each of and , and find binary trees for which they are attained.
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Paper 2, Section II, H
2013 commentDefine a BCH code of length , where is odd, over the field of 2 elements with design distance . Show that the minimum weight of such a code is at least . [Results about the van der Monde determinant may be quoted without proof, provided they are stated clearly.]
Consider a BCH code of length 31 over the field of 2 elements with design distance 8 . Show that the minimum distance is at least 11. [Hint: Let be a primitive element in the field of elements, and consider the minimal polynomial for certain powers of
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Paper 1, Section II, H
2013 commentDefine the bar product of binary linear codes and , where is a subcode of . Relate the rank and minimum distance of to those of and and justify your answer. Show that if denotes the dual code of , then
Using the bar product construction, or otherwise, define the Reed-Muller code for . Show that if , then the dual of is again a Reed-Muller code.
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Paper 4, Section I, D
2013 commentList the relativistic species of bosons and fermions from the standard model of particle physics that are present in the early universe when the temperature falls to .
Which of the particles above will be interacting when the temperature is above and between , respectively?
Explain what happens to the populations of particles present when the temperature falls to .
The entropy density of fermion and boson species with temperature is , where is the number of relativistic spin degrees of freedom, that is,
Show that when the temperature of the universe falls below the ratio of the neutrino and photon temperatures will be given by
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Paper 3, Section I, D
2013 commentThe number densities of protons of mass or neutrons of mass in kinetic equilibrium at temperature , in the absence of any chemical potentials, are each given by (with or )
where is Boltzmann's constant and is the spin degeneracy.
Use this to show, to a very good approximation, that the ratio of the number of neutrons to protons at a temperature is given by
where . Explain any approximations you have used.
The reaction rate for weak interactions between protons and neutrons at energies is given by and the expansion rate of the universe at these energies is given by . Give an example of a weak interaction that can maintain equilibrium abundances of protons and neutrons at these energies. Show how the final abundance of neutrons relative to protons can be calculated and use it to estimate the mass fraction of the universe in helium- 4 after nucleosynthesis.
What would have happened to the helium abundance if the proton and neutron masses had been exactly equal?
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Paper 2, Section I, D
2013 commentThe linearised equation for the growth of small inhomogeneous density perturbations with comoving wavevector in an isotropic and homogeneous universe is
where is the matter density, is the sound speed, is the pressure, is the expansion scale factor of the unperturbed universe, and overdots denote differentiation with respect to time .
Define the Jeans wavenumber and explain its physical meaning.
Assume the unperturbed Friedmann universe has zero curvature and cosmological constant and it contains only zero-pressure matter, so that . Show that the solution for the growth of density perturbations is given by
Comment briefly on the cosmological significance of this result.
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Paper 1, Section I, D
2013 commentThe Friedmann equation and the fluid conservation equation for a closed isotropic and homogeneous cosmology are given by
where the speed of light is set equal to unity, is the gravitational constant, is the expansion scale factor, is the fluid mass density and is the fluid pressure, and overdots denote differentiation with respect to the time coordinate .
If the universe contains only blackbody radiation and defines the zero of time , show that
where is a constant. What is the physical significance of the time ? What is the value of the ratio at the time when the scale factor is largest? Sketch the curve of and identify its geometric shape.
Briefly comment on whether this cosmological model is a good description of the observed universe at any time in its history.
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Paper 3, Section II, D
2013 commentThe contents of a spatially homogeneous and isotropic universe are modelled as a finite mass of pressureless material whose radius evolves from some constant reference radius in proportion to the time-dependent scale factor , with
(i) Show that this motion leads to expansion governed by Hubble's Law. If this universe is expanding, explain why there will be a shift in the frequency of radiation between its emission from a distant object and subsequent reception by an observer. Define the redshift of the observed object in terms of the values of the scale factor at the times of emission and reception.
(ii) The expanding universal mass is given a small rotational perturbation, with angular velocity , and its angular momentum is subsequently conserved. If deviations from spherical expansion can be neglected, show that its linear rotational velocity will fall as , where you should determine the value of . Show that this perturbation will become increasingly insignificant compared to the expansion velocity as the universe expands if .
(iii) A distant cloud of intermingled hydrogen (H) atoms and carbon monoxide (CO) molecules has its redshift determined simultaneously in two ways: by detecting radiation from atomic hydrogen and by detecting radiation from rotational transitions in CO molecules. The ratio of the atomic transition frequency to the CO rotational transition frequency is proportional to , where is the fine structure constant. It is suggested that there may be a small difference in the value of the constant between the times of emission and reception of the radiation from the cloud.
Show that the difference in the redshift values for the cloud, , determined separately by observations of the and transitions, is related to , the difference in values at the times of reception and emission, by
(iv) The universe today contains of its total density in the form of pressureless matter and in the form of a dark energy with constant redshift-independent density. If these are the only two significant constituents of the universe, show that their densities were equal when the scale factor of the universe was approximately equal to of its present value.
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Paper 1, Section II, D
2013 commentA spherically symmetric star of total mass has pressure and mass density , where is the radial distance from its centre. These quantities are related by the equations of hydrostatic equilibrium and mass conservation:
where is the mass inside radius .
By integrating from the centre of the star at , where , to the surface of the star at , where , show that
where is the total gravitational potential energy. Show that
If the surface pressure is negligible and the star is a perfect gas of particles of mass with number density and at temperature , and radiation pressure can be ignored, then show that
where is the mean temperature of the star, which you should define.
Hence, show that the mean temperature of the star satisfies the inequality
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Paper 4, Section II, H
2013 commentDefine what is meant by the geodesic curvature of a regular curve parametrized by arc length on a smooth oriented surface . If is the unit sphere in and is a parametrized geodesic circle of radius , with , justify the fact that .
State the general form of the Gauss-Bonnet theorem with boundary on an oriented surface , explaining briefly the terms which occur.
Let now denote the circular cone given by and , for a fixed choice of with , and with a fixed choice of orientation. Let be a simple closed piecewise regular curve on , with (signed) exterior angles at the vertices (that is, is the angle between limits of tangent directions, with sign determined by the orientation). Suppose furthermore that the smooth segments of are geodesic curves. What possible values can take? Justify your answer.
[You may assume that a simple closed curve in bounds a region which is homeomorphic to a disc. Given another simple closed curve in the interior of this region, you may assume that the two curves bound a region which is homeomorphic to an annulus.]
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Paper 3, Section II, H
2013 commentWe say that a parametrization of a smooth surface is isothermal if the coefficients of the first fundamental form satisfy and , for some smooth non-vanishing function on . For an isothermal parametrization, prove that
where denotes the unit normal vector and the mean curvature, which you may assume is given by the formula
where and are coefficients in the second fundamental form.
Given a parametrization of a surface , we consider the complex valued functions on :
Show that is isothermal if and only if . If is isothermal, show that is a minimal surface if and only if are holomorphic functions of the complex variable
Consider the holomorphic functions on (with complex coordinate on given by
Find a smooth map for which and the defined by (2) satisfy the equations (1). Show furthermore that extends to a smooth map . If is the complex coordinate on , show that
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Paper 2, Section II, H
2013 commentLet be a regular curve parametrized by arc length having nowherevanishing curvature. State the Frenet relations between the tangent, normal and binormal vectors at a point, and their derivatives.
Let be a smooth oriented surface. Define the Gauss map , and show that its derivative at , is self-adjoint. Define the Gaussian curvature of at .
Now suppose that has image in and that its normal curvature is zero for all . Show that the Gaussian curvature of at a point of the curve is , where denotes the torsion of the curve.
If is a standard embedded torus, show that there is a curve on for which the normal curvature vanishes and the Gaussian curvature of is zero at all points of the curve.
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Paper 1, Section II, H
2013 commentFor a smooth map of manifolds, define the concepts of critical point, critical value and regular value.
With the obvious identification of with , and hence also of with , show that the complex-valued polynomial determines a smooth map whose only critical point is at the origin. Hence deduce that is a 4-dimensional manifold, and find the equations of its tangent space at any given point .
Now let be the unit 5 -sphere, defined by . Given a point , by considering the vector or otherwise, show that not all tangent vectors to at are tangent to . Deduce that is a compact three-dimensional manifold.
[Standard results may be quoted without proof if stated carefully.]
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Paper 4, Section I, C
2013 commentConsider the system
What is the Poincaré index of the single fixed point? If there is a closed orbit, why must it enclose the origin?
By writing and for suitable functions and , show that if there is a closed orbit then
Deduce that there is no closed orbit when .
If and and are both , where is a small parameter, then there is a single closed orbit that is to within a circle of radius centred on the origin. Deduce a relation between and .
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Paper 3, Section I, C
2013 commentA one-dimensional map is defined by
where is a parameter. What is the condition for a bifurcation of a fixed point of ?
Let . Find the fixed points and show that bifurcations occur when and . Sketch the bifurcation diagram, showing the locus and stability of the fixed points in the plane and indicating the type of each bifurcation.
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Paper 2, Section I,
2013 commentLet be a two-dimensional dynamical system with a fixed point at . Define a Lyapunov function and explain what it means for to be Lyapunov stable.
For the system
determine the values of for which is a Lyapunov function in a sufficiently small neighbourhood of the origin.
For the case , find and such that at implies that as and at implies that as
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Paper 1, Section I, C
2013 commentConsider the dynamical system in which has a hyperbolic fixed point at the origin.
Define the stable and unstable invariant subspaces of the system linearised about the origin. Give a constraint on the dimensions of these two subspaces.
Define the local stable and unstable manifolds of the origin for the system. How are these related to the invariant subspaces of the linearised system?
For the system
calculate the stable and unstable manifolds of the origin, each correct up to and including cubic order.
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Paper 3, Section II, C
2013 commentLet be a continuous map of an interval . Explain what is meant by the statements (a) has a horseshoe and (b) is chaotic according to Glendinning's definition of chaos.
Assume that has a 3-cycle with , . Prove that has a horseshoe. [You may assume the Intermediate Value Theorem.]
Represent the effect of on the intervals and by means of a directed graph. Explain how the existence of the 3 -cycle corresponds to this graph.
The map has a 4-cycle with , and . If is necessarily chaotic? [You may use a suitable directed graph as part of your argument.]
How does your answer change if ?
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Paper 4, Section II, C
2013 commentConsider the dynamical system
where .
Find the fixed points of the dynamical system. Show that for any fixed value of there exist three values of where a bifurcation occurs. Show that when .
In the remainder of this question set .
(i) Being careful to explain your reasoning, show that the extended centre manifold for the bifurcation at can be written in the form , where and denote the departures from the values of and at the fixed point, and are suitable constants (to be determined) and is quadratic to leading order. Derive a suitable approximate form for , and deduce the nature of the bifurcation and the stability of the different branches of the steady state solution near the bifurcation.
(ii) Repeat the calculations of part (i) for the bifurcation at .
(iii) Sketch the values of the fixed points as functions of , indicating the nature of the bifurcations and where each branch is stable.
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Paper 4, Section II, 35B
2013 comment(i) For a time-dependent source, confined within a domain , show that the time derivative of the dipole moment satisfies
where is the current density.
(ii) The vector potential due to a time-dependent source is given by
where , and is the unit vector in the direction. Calculate the resulting magnetic field . By considering the magnetic field for small show that the dipole moment of the effective source satisfies
Calculate the asymptotic form of the magnetic field at very large .
(iii) Using the equation
calculate at very large . Show that and form a right-handed triad, and moreover . How do and depend on What is the significance of this?
(iv) Calculate the power emitted per unit solid angle and sketch its dependence on . Show that the emitted radiation is polarised and describe how the plane of polarisation (that is, the plane in which and lie) depends on the direction of the dipole. Suppose the dipole moment has constant amplitude and constant frequency and so the radiation is monochromatic with wavelength . How does the emitted power depend on ?
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Paper 3, Section II, 36B
2013 comment(i) Obtain Maxwell's equations in empty space from the action functional
where .
(ii) A modification of Maxwell's equations has the action functional
where again and is a constant. Obtain the equations of motion of this theory and show that they imply .
(iii) Show that the equations of motion derived from admit solutions of the form
where is a constant 4-vector, and the 4 -vector satisfies and .
(iv) Show further that the tensor
is conserved, that is .
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Paper 1, Section II, 36B
2013 comment(i) Starting from
and performing a Lorentz transformation with , using
show how and transform under a Lorentz transformation.
(ii) By taking the limit , obtain the behaviour of and under a Galilei transfomation and verify the invariance under Galilei transformations of the nonrelativistic equation
(iii) Show that Maxwell's equations admit solutions of the form
where is an arbitrary function, is a unit vector, and the constant vectors and are subject to restrictions which should be stated.
(iv) Perform a Galilei transformation of a solution , with . Show that, by a particular choice of , the solution may brought to the form
where is an arbitrary function and is a spatial coordinate in the rest frame. By showing that is not a solution of Maxwell's equations in the boosted frame, conclude that Maxwell's equations are not invariant under Galilei transformations.
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Paper 4, Section II, A
2013 commentConsider the flow of an incompressible fluid of uniform density and dynamic viscosity . Show that the rate of viscous dissipation per unit volume is given by
where is the strain rate.
Determine expressions for and when the flow is irrotational with velocity potential .
In deep water a linearised wave with a surface displacement has a velocity potential . Hence determine the rate of the viscous dissipation, averaged over a wave period , for an irrotational surface wave of wavenumber and small amplitude in a fluid with very small viscosity and great depth .
Calculate the depth-integrated kinetic energy per unit wavelength. Assuming that the average potential energy is equal to the average kinetic energy, show that the total wave energy decreases to leading order as , where should be found.
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Paper 2, Section II, A
2013 commentWrite down the boundary-layer equations for steady two-dimensional flow of a viscous incompressible fluid with velocity outside the boundary layer. Find the boundary layer thickness when , a constant. Show that the boundarylayer equations can be satisfied in this case by a streamfunction with suitable scaling function and similarity variable . Find the equation satisfied by and the associated boundary conditions.
Find the drag on a thin two-dimensional flat plate of finite length placed parallel to a uniform flow. Why does the drag not increase in proportion to the length of the plate? [You may assume that the boundary-layer solution is applicable except in negligibly small regions near the leading and trailing edges. You may also assume that .]
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Paper 3, Section II, A
2013 commentA disk hovers on a cushion of air above an air-table - a fine porous plate through which a constant flux of air is pumped. Let the disk have a radius and a weight and hover at a low height above the air-table. Let the volume flux of air, which has density and viscosity , be per unit surface area. The conditions are such that . Explain the significance of this restriction.
Find the pressure distribution in the air under the disk. Show that this pressure balances the weight of the disk if
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Paper 1, Section II, A
2013 commentThe velocity field and stress tensor satisfy the Stokes equations in a volume bounded by a surface . Let be another solenoidal velocity field. Show that
where and are the strain-rates corresponding to the velocity fields and respectively, and is the unit normal vector out of . Hence, or otherwise, prove the minimum dissipation theorem for Stokes flow.
A particle moves at velocity through a highly viscous fluid of viscosity contained in a stationary vessel. As the particle moves, the fluid exerts a drag force on it. Show that
Consider now the case when the particle is a small cube, with sides of length , moving in a very large vessel. You may assume that
for some constant . Use the minimum dissipation theorem, being careful to declare the domain(s) involved, to show that
[You may assume Stokes' result for the drag on a sphere of radius .]
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Paper 4, Section I, E
2013 commentLet the function be analytic in the upper half-plane and such that as . Show that
where denotes the Cauchy principal value.
Use the Cauchy integral theorem to show that
where and are the real and imaginary parts of .
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Paper 3, Section I, E
2013 commentLet a real-valued function be the real part of a complex-valued function which is analytic in the neighbourhood of a point , where Derive a formula for in terms of and use it to find an analytic function whose real part is
and such that .
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Paper 2, Section I, E
2013 comment(i) Find all branch points of on an extended complex plane.
(ii) Use a branch cut to evaluate the integral
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Paper 1, Section I, E
2013 commentProve that there are no second order linear ordinary homogeneous differential equations for which all points in the extended complex plane are analytic.
Find all such equations which have one regular singular point at .
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Paper 2, Section II, E
2013 commentThe Beta function is defined for as
and by analytic continuation elsewhere in the complex -plane.
Show that:
(i) ;
(ii) .
By considering for all positive integers , deduce that for all with .
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Paper 1, Section II, E
2013 commentShow that the equation
has solutions of the form , where
and the contour is any closed curve in the complex plane, where and are real constants which should be determined.
Use this to find the general solution, evaluating the integrals explicitly.
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Paper 4, Section II, I
2013 comment(i) Let for . For the cases , is it possible to express , starting with integers and using rational functions and (possibly nested) radicals? If it is possible, briefly explain how this is done, assuming standard facts in Galois Theory.
(ii) Let be the rational function field in three variables over , and for integers let be the subfield of consisting of all rational functions in with coefficients in . Show that is Galois, and determine its Galois group. [Hint: For , the map is an automorphism of .]
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Paper 3, Section II, I
2013 commentLet be a prime number and a field of characteristic . Let be the Frobenius map defined by for all .
(i) Prove that is a field automorphism when is a finite field.
(ii) Is the same true for an arbitrary algebraic extension of ? Justify your answer.
(iii) Let be the rational function field in variables where over . Determine the image of , and show that makes into an extension of degree over a subfield isomorphic to . Is it a separable extension?
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Paper 2, Section II, I
2013 commentFor a positive integer , let be the cyclotomic field obtained by adjoining all -th roots of unity to . Let .
(i) Determine the Galois group of over .
(ii) Find all such that is contained in .
(iii) List all quadratic and quartic extensions of which are contained in , in the form or . Indicate which of these fields occurred in (ii).
[Standard facts on the Galois groups of cyclotomic fields and the fundamental theorem of Galois theory may be used freely without proof.]
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Paper 1, Section II, I
2013 comment(i) Give an example of a field , contained in , such that is a product of two irreducible quadratic polynomials in . Justify your answer.
(ii) Let be any extension of degree 3 over . Prove that the polynomial is irreducible over .
(iii) Give an example of a prime number such that is a product of two irreducible quadratic polynomials in . Justify your answer.
[Standard facts on fields, extensions, and finite fields may be quoted without proof, as long as they are stated clearly.]
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Paper 4, Section II, D
2013 commentConsider the metric describing the interior of a star,
defined for by
with
Here , where is the mass of the star, , and we have taken units in which we have set .
(i) The star is made of a perfect fluid with energy-momentum tensor
Here is the 4-velocity of the fluid which is at rest, the density is constant throughout the star and the pressure depends only on the radial coordinate. Write down the Einstein field equations and show that they may be written as
(ii) Using the formulae given below, or otherwise, show that for , one has
where primes denote differentiation with respect to . Hence show that
[The non-zero components of the Ricci tensor are
Note that
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Paper 2, Section II, D
2013 commentA spacetime contains a one-parameter family of geodesics , where is a parameter along each geodesic, and labels the geodesics. The tangent to the geodesics is , and is a connecting vector. Prove that
and hence derive the equation of geodesic deviation:
[You may assume and the Ricci identity in the form
Consider the two-dimensional space consisting of the sphere of radius with line element
Show that one may choose , and that
Hence show that , using the geodesic deviation equation and the identity in any two-dimensional space
where is the Ricci scalar.
Verify your answer by direct computation of .
[You may assume that the only non-zero connection components are
and
You may also use the definition
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Paper 3, Section II, D
2013 commentThe Schwarzschild metric for a spherically symmetric black hole is given by
where we have taken units in which we set . Consider a photon moving within the equatorial plane , along a path with affine parameter . Using a variational principle with Lagrangian
or otherwise, show that
for constants and . Deduce that
Assume now that the photon approaches from infinity. Show that the impact parameter (distance of closest approach) is given by
Denote the right hand side of equation as . By sketching in each of the cases below, or otherwise, show that:
(a) if , the photon is deflected but not captured by the black hole;
(b) if , the photon is captured;
(c) if , the photon orbit has a particular form, which should be described.
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Paper 1, Section II, 37D
2013 commentThe curve , is a geodesic with affine parameter . Write down the geodesic equation satisfied by .
Suppose the parameter is changed to , where . Obtain the corresponding equation and find the condition for to be affine. Deduce that, whatever parametrization is used along the curve , the tangent vector to satisfies
Now consider a spacetime with metric , and conformal transformation
The curve is a geodesic of the metric connection of . What further restriction has to be placed on so that it is also a geodesic of the metric connection of ? Justify your answer.
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Paper 4, Section I,
2013 commentLet be two disjoint closed discs in the Riemann sphere with bounding circles respectively. Let be inversion in the circle and let be the Möbius transformation .
Show that, if , then and so for Deduce that has a fixed point in and a second in .
Deduce that there is a Möbius transformation with
for some .
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Paper 3, Section I,
2013 commentLet be a rank 2 lattice in the Euclidean plane. Show that the group of all Euclidean isometries of the plane that map onto itself is a discrete group. List the possible sizes of the point groups for and give examples to show that point groups of these sizes do arise.
[You may quote any standard results without proof.]
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Paper 2, Section I, G
2013 commentLet be two straight lines in Euclidean 3-space. Show that there is a rotation about some axis through an angle that maps onto . Is this rotation unique?
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Paper 1, Section I, G
2013 commentShow that any pair of lines in hyperbolic 3-space that does not have a common endpoint must have a common normal. Is this still true when the pair of lines does have a common endpoint?
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Paper 1, Section II, G
2013 commentDefine the modular group acting on the upper half-plane.
Describe the set of points in the upper half-plane that have for each . Hence find a fundamental set for acting on the upper half-plane.
Let and be the two Möbius transformations
When is
For any point in the upper half-plane, show that either or else there is an integer with
Deduce that the modular group is generated by and .
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Paper 4, Section II,
2013 commentDefine the limit set for a Kleinian group. If your definition of the limit set requires an arbitrary choice of a base point, you should prove that the limit set does not depend on this choice.
Let be the four discs where is the point respectively. Show that there is a parabolic Möbius transformation that maps the interior of onto the exterior of and fixes the point where and touch. Show further that we can choose so that it maps the unit disc onto itself.
Let be the similar parabolic transformation that maps the interior of onto the exterior of , fixes the point where and touch, and maps the unit disc onto itself. Explain why the group generated by and is a Kleinian group . Find the limit set for the group and justify your answer.
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Paper 4, Section II, F
2013 commentDefine the maximum degree and the chromatic index of the graph .
State and prove Vizing's theorem relating and .
Let be a connected graph such that but, for every subgraph of holds. Show that is a circuit of odd length.
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Paper 3, Section II, F
2013 commentLet be a graph of order and average degree . Let be the adjacency matrix of and let be its characteristic polynomial. Show that and . Show also that is twice the number of triangles in .
The eigenvalues of are . Prove that .
Evaluate . Show that and infer that . Does there exist, for each , a graph with for which ?
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Paper 2, Section II, F
2013 commentLet be a graph with . State and prove a necessary and sufficient condition for to be Eulerian (that is, for to have an Eulerian circuit).
Prove that if then is Hamiltonian (that is, has a Hamiltonian circuit).
The line graph of has vertex set and edge set
Show that is Eulerian if is regular and connected.
Must be Hamiltonian if is Eulerian? Must be Eulerian if is Hamiltonian? Justify your answers.
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Paper 1, Section II, F
2013 commentState and prove Hall's theorem about matchings in bipartite graphs.
Show that a regular bipartite graph has a matching meeting every vertex.
A graph is almost r-regular if each vertex has degree or . Show that, if , an almost -regular graph must contain an almost -regular graph with .
[Hint: First, if possible, remove edges from whilst keeping it almost r-regular.]
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Paper 3, Section II, C
2013 commentLet and be two complex-valued matrix functions, smoothly differentiable in their variables. We wish to explore the solution of the overdetermined linear system
for some twice smoothly differentiable vector function .
Prove that, if the overdetermined system holds, then the functions and obey the zero curvature representation
Let and
where subscripts denote derivatives, is the complex conjugate of and is a constant. Find the compatibility condition on the function so that and obey the zero curvature representation.
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Paper 2, Section II, 32C
2013 commentConsider the Hamiltonian system
where .
When is the transformation canonical?
Prove that, if the transformation is canonical, then the equations in the new variables are also Hamiltonian, with the same Hamiltonian function .
Let , where is a symmetric nonsingular matrix. Determine necessary and sufficient conditions on for the transformation to be canonical.
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Paper 1, Section II, C
2013 commentQuoting carefully all necessary results, use the theory of inverse scattering to derive the 1-soliton solution of the equation
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Paper 3, Section II, F
2013 commentState the Stone-Weierstrass Theorem for real-valued functions.
State Riesz's Lemma.
Let be a compact, Hausdorff space and let be a subalgebra of separating the points of and containing the constant functions. Fix two disjoint, non-empty, closed subsets and of .
(i) If show that there exists such that on , and on . Explain briefly why there is such that on .
(ii) Show that there is an open cover of , elements of and positive integers such that
for each .
(iii) Using the inequality
show that for sufficiently large positive integers , the element
of satisfies
for each .
(iv) Show that the element of satisfies
Now let with . By considering the sets and , show that there exists such that . Deduce that is dense in .
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Paper 4, Section II, F
2013 commentLet be a bounded linear operator on a complex Banach space . Define the spectrum of . What is an approximate eigenvalue of ? What does it mean to say that is compact?
Assume now that is compact. Show that if is in the boundary of and , then is an eigenvalue of . [You may use without proof the result that every in the boundary of is an approximate eigenvalue of .]
Let be a compact Hermitian operator on a complex Hilbert space . Prove the following:
(a) If and , then is an eigenvalue of .
(b) is countable.
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Paper 2, Section II, F
2013 commentLet be a Banach space. Let be a bounded linear operator. Show that there is a bounded sequence in such that for all .
Fix . Define the Banach space and briefly explain why it is separable. Show that for there exists such that and . [You may use Hölder's inequality without proof.]
Deduce that embeds isometrically into .
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Paper 1, Section II, F
2013 commentState and prove the Closed Graph Theorem. [You may assume any version of the Baire Category Theorem provided it is clearly stated. If you use any other result from the course, then you must prove it.]
Let be a closed subspace of such that is also a subset of . Show that the left-shift , given by , is bounded when is equipped with the sup-norm.
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Paper 2, Section II, G
2013 commentExplain what is meant by a chain-complete poset. State the Bourbaki-Witt fixedpoint theorem for such posets.
A poset is called directed if every finite subset of (including the empty subset) has an upper bound in is called directed-complete if every subset of which is directed (in the induced ordering) has a least upper bound in . Show that the set of all chains in an arbitrary poset , ordered by inclusion, is directed-complete.
Given a poset , let denote the set of all order-preserving maps , ordered pointwise (i.e. if and only if for all ). Show that is directed-complete if is.
Now suppose is directed-complete, and that is order-preserving and inflationary. Show that there is a unique smallest set satisfying
(a) ;
(b) is closed under composition (i.e. ); and
(c) is closed under joins of directed subsets.
Show that
(i) all maps in are inflationary;
(ii) is directed;
(iii) if , then all values of are fixed points of ;
(iv) for every , there exists with .
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Paper 3, Section II, G
2013 commentExplain carefully what is meant by syntactic entailment and semantic entailment in the propositional calculus. State the Completeness Theorem for the propositional calculus, and deduce the Compactness Theorem.
Suppose and are pairwise disjoint sets of primitive propositions, and let and be propositional formulae such that is a theorem of the propositional calculus. Consider the set
Show that is inconsistent, and deduce that there exists such that both and are theorems. [Hint: assuming is consistent, take a suitable valuation of and show that
is inconsistent. The Deduction Theorem may be assumed without proof.]
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Paper 4, Section II, G
2013 commentState the Axiom of Foundation and the Principle of -Induction, and show that they are equivalent in the presence of the other axioms of ZF set theory. [You may assume the existence of transitive closures.]
Given a model for all the axioms of ZF except Foundation, show how to define a transitive class which, with the restriction of the given relation , is a model of ZF.
Given a model of , indicate briefly how one may modify the relation so that the resulting structure fails to satisfy Foundation, but satisfies all the other axioms of . [You need not verify that all the other axioms hold in .]
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Paper 1, Section II, G
2013 commentWrite down the recursive definitions of ordinal addition, multiplication and exponentiation.
Given that is a strictly increasing function-class (i.e. implies , show that for all .
Show that every ordinal has a unique representation in the form
where , and .
Under what conditions can an ordinal be represented in the form
where and Justify your answer.
[The laws of ordinal arithmetic (associative, distributive, etc.) may be assumed without proof.]
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Paper 4, Section I, A
2013 commentA model of two populations competing for resources takes the form
where all parameters are positive. Give a brief biological interpretation of and . Briefly describe the dynamics of each population in the absence of the other.
Give conditions for there to exist a steady-state solution with both populations present (that is, and ), and give conditions for this solution to be stable.
In the case where there exists a solution with both populations present but the solution is not stable, what is the likely long-term outcome for the biological system? Explain your answer with the aid of a phase diagram in the plane.
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Paper 3, Section I, A
2013 commentAn immune system creates a burst of new white blood cells with probability per unit time. White blood cells die with probability each per unit time. Write down the master equation for , the probability that there are white blood cells at time .
Given that initially, find an expression for the mean of .
Show that the variance of has the form and find and .
If the immune system were modified to produce times as many cells per burst but with probability per unit time divided by a factor , how would the mean and variance of the number of cells change?
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Paper 2, Section I, A
2013 commentThe population density of individuals of age at time satisfies
with
where is the age-dependent death rate and is the birth rate per individual of age
Seek a similarity solution of the form and show that
Show also that if
then there is such a similarity solution. Give a biological interpretation of .
Suppose now that all births happen at age , at which time an individual produces offspring, and that the death rate is constant with age (i.e. . Find the similarity solution and give the condition for this to represent a growing population.
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Paper 1, Section I, A
2013 commentIn a discrete-time model, a proportion of mature bacteria divides at each time step. When a mature bacterium divides it is destroyed and two new immature bacteria are produced. A proportion of the immature bacteria matures at each time step, and a proportion of mature bacteria dies at each time step. Show that this model may be represented by the equations
Give an expression for the general solution to these equations and show that the population may grow if .
At time , the population is treated with an antibiotic that completely stops bacteria from maturing, but otherwise has no direct effects. Explain what will happen to the population of bacteria afterwards, and give expressions for and for in terms of and .
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Paper 3, Section II, A
2013 commentAn activator-inhibitor system is described by the equations
where .
Find and sketch the range of for which the spatially homogeneous system has a stable stationary solution with and .
Considering spatial perturbations of the form about the solution found above, find conditions for the system to be unstable. Sketch this region in the plane for fixed .
Find , the critical wavenumber at the onset of the instability, in terms of and .
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Paper 2, Section II, A
2013 commentThe concentration of insects at position at time satisfies the nonlinear diffusion equation
with . Find the value of which allows a similarity solution of the form , with .
Show that
where is a constant. From the original partial differential equation, show that the total number of insects does not change in time. From this result, find a general expression relating and . Find a closed-form solution for in the case .
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Paper 4, Section II, H
2013 commentState Dedekind's criterion. Use it to factor the primes up to 5 in the ring of integers of . Show that every ideal in of norm 10 is principal, and compute the class group of .
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Paper 2, Section II, H
2013 comment(i) State Dirichlet's unit theorem.
(ii) Let be a number field. Show that if every conjugate of has absolute value at most 1 then is either zero or a root of unity.
(iii) Let and where . Compute . Show that
Hence or otherwise find fundamental units for and .
[You may assume that the only roots of unity in are powers of ]
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Paper 1, Section II, H
2013 commentLet be a monic irreducible polynomial of degree . Let , where is a root of .
(i) Show that if is square-free then .
(ii) In the case find the minimal polynomial of and hence compute the discriminant of . What is the index of in ?
[Recall that the discriminant of is .]
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Paper 1, Section I, I
2013 commentState and prove Gauss's Lemma for the Legendre symbol . For which odd primes is 2 a quadratic residue modulo Justify your answer.
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Paper 4, Section I, I
2013 commentLet with . Define the Riemann zeta function for . Show that for ,
where the product is taken over all primes. Deduce that there are infinitely many primes.
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Paper 3, Section I, I
2013 commentState the Chinese Remainder Theorem.
A composite number is defined to be a Carmichael number if whenever . Show that a composite is Carmichael if and only if is square-free and divides for all prime factors of . [You may assume that, for an odd prime and an integer, is a cyclic group.]
Show that if with all three factors prime, then is Carmichael.
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Paper 2, Section I, I
2013 commentDefine Euler's totient function , and show that . Hence or otherwise prove that for any prime the multiplicative group is cyclic.
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Paper 4, Section II, I
2013 comment(i) What is meant by the continued fraction expansion of a real number ? Suppose that has continued fraction . Define the convergents to and give the recurrence relations satisfied by the and . Show that the convergents do indeed converge to .
[You need not justify the basic order properties of finite continued fractions.]
(ii) Find two solutions in strictly positive integers to each of the equations
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Paper 3, Section II, I
2013 commentDefine equivalence of binary quadratic forms and show that equivalent forms have the same discriminant.
Show that an integer is properly represented by a binary quadratic form of discriminant if and only if is soluble in integers. Which primes are represented by a form of discriminant ?
What does it mean for a positive definite form to be reduced? Find all reduced forms of discriminant . For each member of your list find the primes less than 100 represented by the form.
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Paper 4, Section II, C
2013 commentConsider the solution of the two-point boundary value problem
with periodic boundary conditions at and . Construct explicitly the linear algebraic system that arises from the application of a spectral method to the above equation.
The Fourier coefficients of are defined by
Prove that the computation of the Fourier coefficients for the truncated system with (where is an even and positive integer, and assuming that outside this range of ) reduces to the solution of a tridiagonal system of algebraic equations, which you should specify.
Explain the term convergence with spectral speed and justify its validity for the derived approximation of .
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Paper 2, Section II, C
2013 commentConsider the advection equation on the unit interval and , where , subject to the initial condition and the boundary condition , where is a given smooth function on .
(i) We commence by discretising the advection equation above with finite differences on the equidistant space-time grid with and . We obtain an equation for that reads
with the condition for all and .
What is the order of approximation (that is, the order of the local error) in space and time of the above discrete solution to the exact solution of the advection equation? Write the scheme in matrix form and deduce for which choices of this approximation converges to the exact solution. State (without proof) any theorems you use. [You may use the fact that for a tridiagonal matrix
the eigenvalues are given by .]
(ii) How does the order change when we replace the central difference approximation of the first derivative in space by forward differences, that is instead of For which choices of is this new scheme convergent?
(iii) Instead of the approximation in (i) we consider the following method for numerically solving the advection equation,
where we additionally assume that is given. What is the order of this method for a fixed ?
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Paper 3, Section II, C
2013 comment(i) Suppose that is a real matrix, and that and are given so that . Further, let be a non-singular matrix such that , where is the first coordinate vector and . Let . Prove that the eigenvalues of are together with the eigenvalues of the bottom right submatrix of .
(ii) Suppose again that is a real matrix, and that two linearly independent vectors are given such that the linear subspace spanned by and is invariant under the action of , that is
Denote by an matrix whose two columns are the vectors and , and let be a non-singular matrix such that is upper triangular, that is
Again, let . Prove that the eigenvalues of are the eigenvalues of the top left submatrix of together with the eigenvalues of the bottom right submatrix of
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Paper 1, Section II, 40C
2013 commentLet
(i) For which values of is positive definite?
(ii) Formulate the Gauss-Seidel method for the solution of a system
with as defined above and . Prove that the Gauss-Seidel method converges to the solution of the above system whenever is positive definite. [You may state and use the Householder-John theorem without proof.]
(iii) For which values of does the Jacobi iteration applied to the solution of the above system converge?
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Paper 4, Section II, K
2013 commentGiven , all positive, it is desired to choose to maximize
subject to .
Explain what Pontryagin's maximum principle guarantees about a solution to this problem.
Show that no matter whether is constrained or unconstrained there is a constant such that the optimal control is of the form . Find an expression for under the constraint .
Show that if is unconstrained then .
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Paper 3, Section II, K
2013 commentA particle follows a discrete-time trajectory in given by
where is a white noise sequence with and . Given , we wish to choose to minimize .
Show that for some this problem can be reduced to one of controlling a scalar state .
Find, in terms of , the optimal . What is the change in minimum achievable when the system starts in as compared to when it starts in ?
Consider now a trajectory starting at . What value of is optimal if we wish to minimize ?
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Paper 2, Section II, K
2013 commentSuppose is a Markov chain. Consider the dynamic programming equation
with , and . Prove that:
(i) is nondecreasing in ;
(ii) , where is the value function of an infinite-horizon problem that you should describe;
(iii) .
A coin lands heads with probability . A statistician wishes to choose between: and , one of which is true. Prior probabilities of and in the ratio change after one toss of the coin to ratio (if the toss was a head) or to ratio (if the toss was a tail). What problem is being addressed by the following dynamic programming equation?
Prove that is a convex function of .
By sketching a graph of , describe the form of the optimal policy.
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Paper 4, Section II, C
2013 comment(i) Show that an arbitrary solution of the one-dimensional wave equation can be written in the form .
Hence, deduce the formula for the solution at arbitrary of the Cauchy problem
where are arbitrary Schwartz functions.
Deduce from this formula a theorem on finite propagation speed for the onedimensional wave equation.
(ii) Define the Fourier transform of a tempered distribution. Compute the Fourier transform of the tempered distribution defined for all by the function
that is, for all . By considering the Fourier transform in , deduce from this the formula for the solution of that you obtained in part (i) in the case .
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Paper 3, Section II, C
2013 commentDefine the parabolic boundary of the domain for .
Let be a smooth real-valued function on which satisfies the inequality
Assume that the coefficients and are smooth functions and that there exist positive constants such that everywhere, and . Prove that
[Here is the positive part of the function .]
Consider a smooth real-valued function on such that
everywhere, and for all . Deduce from that if for all then for all . [Hint: Consider and compute
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Paper 1, Section II, C
2013 comment(i) Discuss briefly the concept of well-posedness of a Cauchy problem for a partial differential equation.
Solve the Cauchy problem
where and denotes the partial derivative with respect to for .
For the case show that the solution satisfies , where the norm on functions of one variable is defined by
Deduce that the Cauchy problem is then well-posed in the uniform metric (i.e. the metric determined by the norm).
(ii) State the Cauchy-Kovalevskaya theorem and deduce that the following Cauchy problem for the Laplace equation,
has a unique analytic solution in some neighbourhood of for any analytic function . Write down the solution for the case , and hence give a sequence of initial data with the property that
whereas , the corresponding solution of , satisfies
for any .
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Paper 2, Section II, C
2013 commentState the Lax-Milgram lemma.
Let be a smooth vector field which is -periodic in each coordinate for . Write down the definition of a weak solution for the equation
to be solved for given in , with both and also -periodic in each co-ordinate. [In this question use the definition
for the Sobolev spaces of functions -periodic in each coordinate and for
If the vector field is divergence-free, prove that there exists a unique weak solution for all such .
Supposing that is the constant vector field with components , write down the solution of in terms of Fourier series and show that there exists such that
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Paper 4, Section II, E
2013 comment(i) The creation and annihilation operators for a harmonic oscillator of angular frequency satisfy the commutation relation . Write down an expression for the Hamiltonian and number operator in terms of and . Explain how the space of eigenstates , of is formed, and deduce the eigenenergies for these states. Show that
(ii) The operator is defined to be
for Show that commutes with . Show that if , then
and otherwise. By considering the action of on the state , deduce that
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Paper 3, Section II, E
2013 commentA particle moves in one dimension in an infinite square-well potential for and for . Find the energy eigenstates. Show that the energy eigenvalues are given by for integer , where is a constant which you should find.
The system is perturbed by the potential . Show that the energy of the level remains unchanged to first order in . Show that the ground-state wavefunction is
where and are numerical constants which you should find. Briefly comment on the conservation of parity in the unperturbed and perturbed systems.
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Paper 2, Section II, 33E
2013 comment(i) In units where , angular momentum states obey
Use the algebra of angular momentum to derive the following in terms of and : (a) ; (b) ; (c) .
(ii) Find in terms of and . Thus calculate the quantum numbers of the state in terms of and . Derive the normalisation of the state . Therefore, show that
finding in terms of .
(iii) Consider the combination of a spinless particle with an electron of spin and orbital angular momentum 1. Calculate the probability that the electron has a spin of in the -direction if the combined system has an angular momentum of in the -direction and a total angular momentum of . Repeat the calculation for a total angular momentum of .
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Paper 1, Section II, E
2013 commentConsider a composite system of several identical particles. Describe how the multiparticle state is constructed from single-particle states. For the case of two identical particles, describe how considering the interchange symmetry leads to the definition of bosons and fermions.
Consider two non-interacting, identical particles, each with spin 1 . The singleparticle, spin-independent Hamiltonian has non-degenerate eigenvalues and wavefunctions where labels the particle and In terms of these single-particle wavefunctions and single-particle spin states and , write down all of the two-particle states and energies for:
(i) the ground state;
(ii) the first excited state.
Assume now that is a linear function of . Find the degeneracy of the energy level of the two-particle system for:
(iii) even;
(iv) odd.
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Paper 4, Section II,
2013 commentAssuming only the existence and properties of the univariate normal distribution, define , the multivariate normal distribution with mean (row-)vector and dispersion matrix ; and , the Wishart distribution on integer degrees of freedom and with scale parameter . Show that, if , and are fixed, then , where .
The random matrix has rows that are independently distributed as , where both parameters and are unknown. Let , where 1 is the vector of ; and , with . State the joint distribution of and given the parameters.
Now suppose and is positive definite. Hotelling's is defined as
where with . Show that, for any values of and ,
the distribution on and degrees of freedom.
[You may assume that:
- If and is a fixed vector, then
- If are independent, then
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Paper 3, Section II, K
2013 commentWhat is meant by a convex decision problem? State and prove a theorem to the effect that, in a convex decision problem, there is no point in randomising. [You may use standard terms without defining them.]
The sample space, parameter space and action space are each the two-point set . The observable takes value 1 with probability when the parameter , and with probability when . The loss function is 0 if , otherwise 1 . Describe all the non-randomised decision rules, compute their risk functions, and plot these as points in the unit square. Identify an inadmissible non-randomised decision rule, and a decision rule that dominates it.
Show that the minimax rule has risk function , and is Bayes against a prior distribution that you should specify. What is its Bayes risk? Would a Bayesian with this prior distribution be bound to use the minimax rule?
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Paper 1, Section II, K
2013 commentWhen the real parameter takes value , variables arise independently from a distribution having density function with respect to an underlying measure . Define the score variable and the information function for estimation of based on , and relate to .
State and prove the Cramér-Rao inequality for the variance of an unbiased estimator of . Under what conditions does this inequality become an equality? What is the form of the estimator in this case? [You may assume , and any further required regularity conditions, without comment.]
Let be the maximum likelihood estimator of based on . What is the asymptotic distribution of when ?
Suppose that, for each is unbiased for , and the variance of is exactly equal to its asymptotic variance. By considering the estimator , or otherwise, show that, for .
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Paper 2, Section II, K
2013 commentDescribe the Weak Sufficiency Principle (WSP) and the Strong Sufficiency Principle (SSP). Show that Bayesian inference with a fixed prior distribution respects WSP.
A parameter has a prior distribution which is normal with mean 0 and precision (inverse variance) Given , further parameters have independent normal distributions with mean and precision . Finally, given both and , observables are independent, being normal with mean , and precision . The precision parameters are all fixed and known. Let , where . Show, directly from the definition of sufficiency, that is sufficient for . [You may assume without proof that, if have independent normal distributions with the same variance, and , then the vector is independent of .]
For data-values , determine the joint distribution, say, of , given and What is the distribution of , given and
Using these results, describe clearly how Gibbs sampling combined with RaoBlackwellisation could be applied to estimate the posterior joint distribution of , given .
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Paper 4, Section II, K
2013 commentState Birkhoff's almost-everywhere ergodic theorem.
Let be a sequence of independent random variables such that
Define for
What is the distribution of Show that the random variables and are not independent.
Set . Show that converges as almost surely and determine the limit. [You may use without proof any standard theorem provided you state it clearly.]
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Paper 3, Section II,
2013 commentLet be an integrable random variable with . Show that the characteristic function is differentiable with . [You may use without proof standard convergence results for integrals provided you state them clearly.]
Let be a sequence of independent random variables, all having the same distribution as . Set . Show that in distribution. Deduce that in probability. [You may not use the Strong Law of Large Numbers.]
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Paper 2, Section II,
2013 commentLet be a sequence of non-negative measurable functions defined on a measure space . Show that is also a non-negative measurable function.
State the Monotone Convergence Theorem.
State and prove Fatou's Lemma.
Let be as above. Suppose that as for all . Show that
Deduce that, if is integrable and , then converges to in . [Still assume that and are as above.]
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Paper 1, Section II,
2013 commentState Dynkin's -system -system lemma.
Let and be probability measures on a measurable space . Let be a -system on generating . Suppose that for all . Show that .
What does it mean to say that a sequence of random variables is independent?
Let be a sequence of independent random variables, all uniformly distributed on . Let be another random variable, independent of . Define random variables in by . What is the distribution of ? Justify your answer.
Show that the sequence of random variables is independent.
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Paper 3, Section II, G
2013 commentSuppose that and are complex representations of the finite groups and respectively. Use and to construct a representation of on and show that its character satisfies
for each .
Prove that if and are irreducible then is irreducible as a representation of . Moreover, show that every irreducible complex representation of arises in this way.
Is it true that every complex representation of is of the form with a complex representation of for Justify your answer.
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Paper 2, Section II, G
2013 commentRecall that a regular icosahedron has 20 faces, 30 edges and 12 vertices. Let be the group of rotational symmetries of a regular icosahedron.
Compute the conjugacy classes of . Hence, or otherwise, construct the character table of . Using the character table explain why must be a simple group.
[You may use any general theorems provided that you state them clearly.]
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Paper 4, Section II, G
2013 commentState and prove Burnside's -theorem.
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Paper 1, Section II, 19G
2013 commentState and prove Maschke's Theorem for complex representations of finite groups.
Without using character theory, show that every irreducible complex representation of the dihedral group of order , has dimension at most two. List the irreducible complex representations of up to isomorphism.
Let be the set of vertices of a regular pentagon with the usual action of . Explicitly decompose the permutation representation into a direct sum of irreducible subrepresentations.
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Paper 3, Section II, I
2013 commentLet be a lattice in where , and let be the complex torus
(i) Give the definition of an elliptic function with respect to . Show that there is a bijection between the set of elliptic functions with respect to and the set of holomorphic maps from to the Riemann sphere. Next, show that if is an elliptic function with respect to and , then is constant.
(ii) Assume that
defines a meromorphic function on , where the sum converges uniformly on compact subsets of . Show that is an elliptic function with respect to . Calculate the order of .
Let be an elliptic function with respect to on , which is holomorphic on and whose only zeroes in the closed parallelogram with vertices are simple zeroes at the points . Show that is a non-zero constant multiple of .
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Paper 2, Section II, I
2013 comment(i) Show that the open unit is biholomorphic to the upper half-plane .
(ii) Define the degree of a non-constant holomorphic map between compact connected Riemann surfaces. State the Riemann-Hurwitz formula without proof. Now let be a complex torus and a holomorphic map of degree 2 , where is the Riemann sphere. Show that has exactly four branch points.
(iii) List without proof those Riemann surfaces whose universal cover is the Riemann sphere or . Now let be a holomorphic map such that there are two distinct elements outside the image of . Assuming the uniformization theorem and the monodromy theorem, show that is constant.
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Paper 1, Section II, I
2013 comment(i) Let be a power series with radius of convergence in . Show that there is at least one point on the circle which is a singular point of , that is, there is no direct analytic continuation of in any neighbourhood of .
(ii) Let and be connected Riemann surfaces. Define the space of germs of function elements of into . Define the natural topology on and the natural . [You may assume without proof that the topology on is Hausdorff.] Show that is continuous. Define the natural complex structure on which makes it into a Riemann surface. Finally, show that there is a bijection between the connected components of and the complete holomorphic functions of into .
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Paper 4, Section I, J
2013 commentThe output of a process depends on the levels of two adjustable variables: , a factor with four levels, and , a factor with two levels. For each combination of a level of and a level of , nine independent values of are observed.
Explain and interpret the commands and (abbreviated) output below. In particular, describe the model being fitted, and describe and comment on the hypothesis tests performed under the summary and anova commands.

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Paper 3, Section I, J
2013 commentConsider the linear model where , and , with independent random variables. The matrix is known and is of full rank . Give expressions for the maximum likelihood estimators and of and respectively, and state their joint distribution. Show that is unbiased whereas is biased.
Suppose that a new variable is to be observed, satisfying the relationship
where is known, and independently of . We propose to predict by . Identify the distribution of
where
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Paper 2, Section I, J
2013 commentConsider a linear model , where and are with , is , and is of full . Let and be sub-vectors of . What is meant by orthogonality between and ?
Now suppose
where are independent random variables, are real-valued known explanatory variables, and is a cubic polynomial chosen so that is orthogonal to and is orthogonal to .
Let . Describe the matrix such that . Show that is block diagonal. Assuming further that this matrix is non-singular, show that the least-squares estimators of and are, respectively,
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Paper 1, Section I, J
2013 commentVariables are independent, with having a density governed by an unknown parameter . Define the deviance for a model that imposes relationships between the .
From this point on, suppose . Write down the log-likelihood of data as a function of .
Let be the maximum likelihood estimate of under model . Show that the deviance for this model is given by
Now suppose that, under , where are known -dimensional explanatory variables and is an unknown -dimensional parameter. Show that satisfies , where and is the matrix with rows , and express this as an equation for the maximum likelihood estimate of . [You are not required to solve this equation.]
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Paper 4, Section II, J
2013 commentLet be a probability density function, with cumulant generating function . Define what it means for a random variable to have a model function of exponential dispersion family form, generated by .
A random variable is said to have an inverse Gaussian distribution, with parameters and (both positive), if its density function is
Show that the family of all inverse Gaussian distributions for is of exponential dispersion family form. Deduce directly the corresponding expressions for and in terms of and . What are the corresponding canonical link function and variance function?
Consider a generalized linear model, , for independent variables , whose random component is defined by the inverse Gaussian distribution with link function thus , where is the vector of unknown regression coefficients and is the vector of known values of the explanatory variables for the observation. The vectors are linearly independent. Assuming that the dispersion parameter is known, obtain expressions for the score function and Fisher information matrix for . Explain how these can be used to compute the maximum likelihood estimate of .
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Paper 1, Section II, J
2013 commentA cricket ball manufacturing company conducts the following experiment. Every day, a bowling machine is set to one of three levels, "Medium", "Fast" or "Spin", and then bowls 100 balls towards the stumps. The number of times the ball hits the stumps and the average wind speed (in kilometres per hour) during the experiment are recorded, yielding the following data (abbreviated):
Write down a reasonable model for , where is the number of times the ball hits the stumps on the day. Explain briefly why we might want to include interactions between the variables. Write code to fit your model.
The company's statistician fitted her own generalized linear model using , and obtained the following summary (abbreviated):
Why are LevelMedium and Wind: LevelMedium not listed?
Suppose that, on another day, the bowling machine is set to "Spin", and the wind speed is 5 kilometres per hour. What linear function of the parameters should the statistician use in constructing a predictor of the number of times the ball hits the stumps that day?
Based on the above output, how might you improve the model? How could you fit your new model in ?
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Paper 4, Section II, A
2013 commentA classical particle of mass moving non-relativistically in two-dimensional space is enclosed inside a circle of radius and attached by a spring with constant to the centre of the circle. The particle thus moves in a potential
where . Let the particle be coupled to a heat reservoir at temperature .
(i) Which of the ensembles of statistical physics should be used to model the system?
(ii) Calculate the partition function for the particle.
(iii) Calculate the average energy and the average potential energy of the particle.
(iv) What is the average energy in:
(a) the limit (strong coupling)?
(b) the limit (weak coupling)?
Compare the two results with the values expected from equipartition of energy.
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Paper 3, Section II, 35A
2013 comment(i) Briefly describe the microcanonical ensemble.
(ii) For quantum mechanical systems the energy levels are discrete. Explain why we can write the probability distribution in this case as
What assumption do we make for the energy interval ?
Consider independent linear harmonic oscillators of equal frequency . Their total energy is given by
Here is the excitation number of oscillator .
(iii) Show that, for fixed and , the number of possibilities to distribute the excitations over oscillators (i.e. the number of different choices consistent with ) is given by
[Hint: You may wish to consider the set of oscillators plus "additional" excitations and what it means to choose objects from this set.]
(iv) Using the probability distribution of part (ii), calculate the probability distribution for the "first" oscillator as a function of its energy .
(v) If then exactly one value of will correspond to a total energy inside the interval . In this case, show that
Approximate this result in the limit .
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Paper 2, Section II, 35A
2013 comment(i) The first law of thermodynamics is , where is the chemical potential. Briefly describe its meaning.
(ii) What is equipartition of energy? Under which conditions is it valid? Write down the heat capacity at constant volume for a monatomic ideal gas.
(iii) Starting from the first law of thermodynamics, and using the fact that for an ideal gas , show that the entropy of an ideal gas containing particles can be written as
where and are temperature and volume of the gas, is the Boltzmann constant, and we define the heat capacity per particle as .
(iv) The Gibbs free energy is defined as . Verify that it is a function of temperature , pressure and particle number . Explain why depends on the particle number through .
(v) Calculate the chemical potential for an ideal gas with heat capacity per particle . Calculate for the special case of a monatomic gas.
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Paper 1, Section II, 35A
2013 comment(i) What is the occupation number of a state with energy according to the Fermi-Dirac statistics for a given chemical potential ?
(ii) Assuming that the energy is spin independent, what is the number of electrons which can occupy an energy level?
(iii) Consider a semi-infinite metal slab occupying (and idealized to have infinite extent in the plane) and a vacuum environment at . An electron with momentum inside the slab will escape the metal in the direction if it has a sufficiently large momentum to overcome a potential barrier relative to the Fermi energy , i.e. if
where is the electron mass.
At fixed temperature , some fraction of electrons will satisfy this condition, which results in a current density in the direction (an electron having escaped the metal once is considered lost, never to return). Each electron escaping provides a contribution to this current density, where is the velocity and the elementary charge.
(a) Briefly describe the Fermi-Dirac distribution as a function of energy in the limit , where is the Boltzmann constant. What is the chemical potential in this limit?
(b) Assume that the electrons behave like an ideal, non-relativistic Fermi gas and that and . Calculate the current density associated with the electrons escaping the metal in the direction. How could we easily increase the strength of the current?
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Paper 4, Section II, J
2013 commentLet denote the time- prices of risky assets in which an agent may invest, . He may also invest his money in a bank account, which will return interest at rate . At time 0 , he knows and , and he knows that . If he chooses at time 0 to invest cash value in risky asset , express his wealth at time 1 in terms of his initial wealth , the choices , the value of , and .
Suppose that his goal is to minimize the variance of subject to the requirement that the mean should be at least , where is given. What portfolio should he choose to achieve this?
Suppose instead that his goal is to minimize subject to the same constraint. Show that his optimal portfolio is unchanged.
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Paper 3, Section II, J
2013 commentSuppose that is a sequence of independent and identically distributed random variables such that for all . Each day, an agent receives an income, the income on day being . After receiving this income, his wealth is . From this wealth, he chooses to consume , and invests the remainder in a bank account which pays a daily interest rate of . Write down the equation for the evolution of .
Suppose we are given constants , and define the functions
where . The agent's objective is to attain
where the supremum is taken over all adapted sequences . If the value function is defined for by
with , explain briefly why you expect the to satisfy
Show that the solution to has the form
for constants to be identified. What is the form of the consumption choices that achieve the supremum in ?
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Paper 1, Section II, 29J
2013 comment(i) Suppose that the price of an asset at time is given by
where is a Brownian motion, and are positive constants, and is the riskless rate of interest, assumed constant. In this model, explain briefly why the time-0 price of a derivative which delivers a bounded random variable at time should be given by . What feature of this model ensures that the price is unique?
Derive an expression for the time- 0 price of a European call option with strike and expiry . Explain the italicized terms.
(ii) Suppose now that the price of an asset at time is given by
where the and are positive constants, and the other notation is as in part (i) above. Show that the time-0 price of a European call option with strike and expiry written on this asset can be expressed as
where the are constants. Explain how the are characterized.
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Paper 2, Section II, J
2013 commentWhat does it mean to say that is a supermartingale?
State and prove Doob's Upcrossing Inequality for a supermartingale.
Let be a martingale indexed by negative time, that is, for each , and . Using Doob's Upcrossing Inequality, prove that the limit exists almost surely.
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Paper 4, Section I, F
2013 commentState the Baire Category Theorem. A set is said to be a -set if it is the intersection of countably many open sets. Show that the set of rationals is not a -set.
[You may assume that the rationals are countable and that is complete.]
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Paper 3, Section I,
2013 commentState Brouwer's fixed point theorem. Let be a continuous function with the property that for all . Show that is surjective.
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Paper 2, Section I, F
2013 comment(i) Show that for every there is a polynomial such that for all satisfying .
[You may assume standard results provided they are stated clearly.]
(ii) Show that there is no polynomial such that for all satisfying .
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Paper 1, Section I,
2013 commentShow that is irrational. [The angle is measured in radians.]
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Paper 2, Section II, F
2013 comment(i) Let be an integer. Show that
(ii) Let us say that an irrational number is badly approximable if there is some constant such that
for all and for all integers . Show that if the integers in the continued fraction expansion are bounded then is badly approximable.
Give, with proof, an example of an irrational number which is not badly approximable.
[Standard facts about continued fractions may be used without proof provided they are stated clearly.]
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Paper 3, Section II, F
2013 commentSuppose that are distinct points. Let be an infinitely differentiable real-valued function on an open interval containing . Let be the unique polynomial of degree at most such that for . Show that for each there is some such that
Now take . Show that
for all . Deduce that there is a polynomial of degree at most such that
for all .
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Paper 4, Section II, 38C
2013 commentA wave disturbance satisfies the equation
where is a positive constant. Find the dispersion relation, and write down the solution to the initial-value problem for which for all , and is given in the form
where is a real function with , so that is real and even.
Use the method of stationary phase to obtain an approximation to for large , with taking the constant value , and . Explain briefly why your answer is inappropriate if .
[You are given that
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Paper 2, Section II, C
2013 commentShow that the equations governing linear elasticity have plane-wave solutions, distinguishing between and waves.
A semi-infinite elastic medium in (where is the vertical coordinate) with density and Lamé moduli and is overlaid by a layer of thickness in of a second elastic medium with density and Lamé moduli and . The top surface at is free, that is, the surface tractions vanish there. The speed of the S-waves is lower in the layer, that is, . For a time-harmonic SH-wave with horizontal wavenumber and frequency , which oscillates in the slow top layer and decays exponentially into the fast semi-infinite medium, derive the dispersion relation for the apparent horizontal wave speed :
Show graphically that for a given value of there is always at least one real value of which satisfies equation . Show further that there are one or more higher modes if
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Paper 3, Section II, C
2013 commentThe dispersion relation for sound waves of frequency in a stationary homogeneous gas is , where is the speed of sound and is the wavenumber. Derive the dispersion relation for sound waves of frequency in a uniform flow with velocity U.
For a slowly-varying medium with local dispersion relation , derive the ray-tracing equations
explaining carefully the meaning of the notation used.
Suppose that two-dimensional sound waves with initial wavenumber are generated at the origin in a gas occupying the half-space . If the gas has a slowlyvarying mean velocity , where , show:
(a) that if and the waves reach a maximum height (which should be identified), and then return to the level in a finite time;
(b) that if and then there is no bound on the height to which the waves propagate.
Comment briefly on the existence, or otherwise, of a quiet zone.
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Paper 1, Section II, 39C
2013 commentStarting from the equations for the one-dimensional unsteady flow of a perfect gas of uniform entropy, show that the Riemann invariants
are constant on characteristics given by , where is the velocity of the gas, is the local speed of sound, is a constant and is the ratio of specific heats.
Such a gas initially occupies the region to the right of a piston in an infinitely long tube. The gas and the piston are initially at rest with . At time the piston starts moving to the left at a constant velocity . Find and in the three regions
where . What is the largest value of for which is positive throughout region (iii)? What happens if exceeds this value?
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Paper 3, Section I, A
2013 commentA cylindrical drinking cup has thin curved sides with density per unit area, and a disk-shaped base with density per unit area. The cup has capacity to hold a fixed volume of liquid. Use the method of Lagrange multipliers to find the minimum mass of the cup.
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Paper 2, Section II, A
2013 commentStarting from the Euler-Lagrange equation, show that a condition for
to be stationary is
In the half-plane , light has speed where . Find the equation for a light ray between and . Sketch the solution.
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Paper 4, Section II, A
2013 commentDerive the Euler-Lagrange equation for the integral
where prime denotes differentiation with respect to , and both and are specified at .
Find that extremises the integral
subject to and .
Show that your solution is a global maximum. You may use the result that
for any (suitably differentiable) function which satisfies and .